收益率曲線交易的分析框架

目錄

• A Framework for Analyzing Yield Curve Trades
  • INTRODUCTION
  • FORWARD RATES AND THEIR DETERMINANTS
    • How Do the Main Determinants Influence the Yield Curve Shape?
    • Decomposing Forward Rates Into Their Main Determinants
  • DECOMPOSING EXPECTED RETURNS OF BOND POSITIONS
    • Five Alternative Expected Return Measures
    • Link to Scenario Analysis
  • APPENDIX A. DECOMPOSING THE FORWARD RATE STRUCTURE INTO ITS MAIN DETERMINANTS
  • APPENDIX B. RELATING VARIOUS STATEMENTS ABOUT FORWARD RATES TO EACH OTHER

著：Antti Ilmanen 譯：徐瑞龍

A Framework for Analyzing Yield Curve Trades

收益率曲線交易的分析框架

INTRODUCTION

引言

In Part 1 of this series, Overview of Forward Rate Analysis, we argued that the shape of the yield curve depends on three factors: the market's rate expectations; the required bond risk premia; and the convexity bias. After examining these determinants in detail in Parts 2-5, we now return to the "big picture" to show how we can decompose the forward rate curve into these three determinants. Even though we cannot directly observe these determinants, the decomposition can clarify our thinking about the yield curve.

在本系列的第1部分（《遠期收益率分析概述》）中，咱們認爲收益率曲線的形狀取決於三個因素：市場收益率預期、債券風險溢價和凸度誤差。在第2-5部分詳細研究了這些決定因素後，咱們如今回到「大局」上，顯示如何將遠期收益率曲線分解爲這三個決定因素。即便咱們不能直接觀察這些決定因素，分解能夠理清咱們對收益率曲線的思考。

Our analysis also produces direct applications —— it provides a systematic framework for relative value analysis of noncallable government bonds. Analogous to the decomposition of forward rates, the total expected return of any government bond position can be viewed as the sum of a few simple building blocks: (1) the yield income; (2) the rolldown return; (3) the value of convexity; and (4) the duration impact of the rate view. A fifth term, the financing advantage, should be added for bonds that trade "special" in the repo market.

咱們的分析也產生了直接的應用，它提供了一個系統的框架，用於非可贖回政府債券的相對價值分析。相似於遠期收益率的分解，任何政府債券頭寸的總預期回報能夠被看做是幾個簡單因素的總和：（1）收益率回報；（2）下滑回報；（3）凸度價值和（4）久期影響。還有第五個因素——融資優點，這是在回購市場上被「特殊」對待的債券所具備的。

The following observations motivate this decomposition. A bond's near-term expected return is a sum of its horizon return given an unchanged yield curve and its expected return from expected changes in the yield curve. The first item, the horizon return, is also called the rolling yield because it is a sum of the bond's yield income and the rolldown return (the capital gain that the bond earns because its yield declines as its maturity shortens and it "rolls down" an upward-sloping yield curve). The second item, the expected return from expected changes in the yield curve, can be approximated by duration and convexity effects. The duration impact is zero if the yield curve is expected to remain unchanged, but it may be the main source of expected return if the rate predictions are based on a subjective market view or on a quantitative forecasting model. The value of convexity is always positive and depends on the bonds convexity and on the perceived level of yield volatility.

如下觀察結果啓發了這種分解。債券的近期預期回報是收益率曲線不變狀況下的持有期回報及收益率曲線預期變化帶來的預期回報的和。第一個部分，即持有期回報，也被稱爲滾動收益率，由於它是債券收益率和下滑回報（在向上傾斜的收益率曲線上收益率隨着其期限的縮短而降低，收益率在曲線上「下滑」而使得債券賺取的資本回報）的總和。第二個部分，收益率曲線預期變化帶來的預期回報，能夠由久期和凸度來近似。若是預期收益率曲線保持不變，則久期影響爲零，但久期影響多是預期回報的主要來源，若是收益率預測基於主觀的市場觀點或量化預測模型。凸度價值老是爲正的，並取決於債券的凸度和收益率波動率的水平。

We argue that both prospective and historical relative value analysis should focus on near-term expected return differentials across bond positions instead of on yield spreads. The former measures take into account all sources of expected return. Moreover, they provide a consistent framework for evaluating all types of government bond positions. We also show, with practical examples, how various expected return measures are computed and how our framework for relative value analysis is related to the better-known scenario analysis.

咱們認爲，前瞻性和歷史性的相對價值分析都應該關注債券頭寸的近期預期回報差別，而不是利差。前者考慮到全部的預期回報來源。此外，近期預期回報差別爲評估各種政府債券頭寸提供了一個一致的框架。咱們還經過實際的例子，展現了各類預期回報的計算方法，以及咱們的相對價值分析框架如何與更爲衆所周知的情景分析相關。

FORWARD RATES AND THEIR DETERMINANTS

遠期收益率及其決定因素

How Do the Main Determinants Influence the Yield Curve Shape?

主要決定因素如何影響收益率曲線形狀

We first describe how the market's rate expectations, the required bond risk premia¹, and the convexity bias influence the term structure of spot and forward rates. The market's expectations regarding the future interest rate behavior are probably the most important influences on today's term structure. Expectations for parallel increases in yields tend to make today's term structure linearly upward sloping, and expectations for falling yields tend to make today's term structure inverted. Expectations for future curve flattening induce today's spot and forward rate curves to be concave (functions of maturity), and expectations for future curve steepening induce today's spot and forward rate curves to be convex.² These are the facts, but what is the intuition behind these relationships?

咱們首先描述市場的收益率預期、債券風險溢價以及凸度誤差如何影響即期和遠期收益率的期限結構。市場對將來收益率行爲的預期多是對當下期限結構最重要的影響因素。收益率水平上升的預期傾向於使當下的期限結構呈線性向上傾斜，對收益率降低的預期傾向於使當下的期限結構倒掛。對將來曲線變平的預期會致使當下的即期和遠期收益率曲線爲上凸的（做爲期限的函數），對將來曲線變陡的預期會致使當下的即期和遠期收益率曲線是下凸的。這些是事實，但這些關係背後的直覺是什麼？

The traditional intuition is based on the pure expectations hypothesis. In the absence of risk premia and convexity bias, a long rate is a weighted average of the expected short rates over the life of the long bond. If the short rates are expected to rise, the expected average future short rate (that is, the long rate) is higher than the current short rate, making today's term structure upward sloping. A similar logic explains why expectations of falling rates make today's term structure inverted. However, this logic gives few insights about the relation between the market's expectations regarding future curve reshaping and the curvature of today's term structure.

傳統上，直覺基於徹底預期假說。在沒有風險溢價和凸度誤差的狀況下，長期收益率是長期債券存續期內預期的短時間收益率的加權平均值。若是預期短時間收益率上漲，預期的將來平均短時間收益率（即長期收益率）高於目前的短時間收益率，使得當下的期限結構向上傾斜。相似的邏輯解釋了爲何對預期收益率降低使當下的期限結構倒掛。然而，這個邏輯對於解釋市場對將來曲線形變的預期與當下期限結構的曲率之間的關係幾乎沒有幫助。

Another perspective to the pure expectations hypothesis may provide a better intuition. The absence of risk premia means that all bonds, independent of maturity, have the same near-term expected return. Recall that a bond's holding-period return equals the sum of the initial yield and the capital gains/losses that yield changes cause. Therefore, if all bonds are to have the same expected return, initial yield differentials across bonds must offset any expected capital gains/losses. Similarly, each bond portfolio with expected capital gains must have a yield disadvantage relative to the riskless asset. If investors expect the long bonds to gain value because of a decline in interest rates, they accept a lower initial yield for long bonds than for short bonds, making today's spot and forward rate curves inverted. Conversely, if investors expect the long bonds to lose value because of an increase in interest rates, they demand a higher initial yield for long bonds than for short bonds, making today's spot and forward rate curves upward sloping. Similarly, if investors expect the curve-flattening positions to earn capital gains because of future curve flattening, they accept a lower initial yield for these positions. In such a case, barbells would have lower yields than duration-matched bullets (to equate their near-term expected returns), making today's spot and forward rate curves concave. A converse logic links the market's curve-steepening expectations to convex spot and forward rate curves.

徹底預期假說的另外一個觀點可能會提供更好的直覺。沒有風險溢價意味着，不管期限多少，全部債券具備相同的近期預期回報。回想一下，債券的持有期回報等於初始收益率和收益率變化引發的資本損益的總和。所以，若是全部債券都具備相同的預期回報，則債券之間的初始收益率差別必須抵消任何預期的資本損益。一樣，每一個具備預期資本回報的債券組合都必須相對於無風險資產的存在收益率上的劣勢。若是投資者預期長期債券因爲收益率降低而得到價值，則他們接受長期債券的初始收益率低於短時間債券，使得當下的即期和遠期收益率曲線倒掛。相反，若是投資者預期長期債券因爲收益率上升而失去價值，則他們要求長期債券的初始收益率高於短時間債券，使得當下的即期和遠期收益率曲線向上傾斜。一樣，若是投資者預期作平曲線的頭寸因爲將來曲線變平而得到資本回報，他們接受這些頭寸的初始收益率較低。在這種狀況下，槓鈴組合的收益率將低於久期匹配的子彈組合（使得近期預期回報相等），使當下的即期和遠期收益率曲線變得上凸。相反的邏輯將市場的曲線變陡的預期與即期和遠期收益率曲線下凸聯繫起來。

The above analysis presumes that all bond positions have the same near-term expected returns. In reality, investors require higher returns for holding long bonds than short bonds. Many models that acknowledge bond risk premia assume that they increase linearly with duration (or with return volatility) and that they are constant over time. Parts 3 and 4 of this series showed that empirical evidence contradicts both assumptions. Historical average returns increase substantially with duration at the front end of the curve but only marginally after the two-year duration. Thus, the bond risk premia make the term structure upward-sloping and concave, on average. Moreover, it is possible to forecast when the required bond risk premia are abnormally high or low. Thus, the time-variation in the bond risk premia can cause significant variation in the shape of the term structure.

上述分析假設全部債券頭寸都具備相同的近期預期回報。實際上，投資者要求持有長期債券的回報比短時間債券要高。許多模型接受債券風險溢價的存在，而且假設它們隨久期（或回報波動率）線性增長以及隨時間恆定。本系列的第3和第4部分顯示，實證證據與兩種假設相矛盾。歷史平均回報在曲線前端關於久期大幅度增加，但在兩年久期後只是略有增加。所以，債券風險溢價使得期限結構向上傾斜並上凸。此外，有可能預測債券風險溢價什麼時候異常高或低。所以，債券風險溢價的時變性可能致使期限結構形狀的顯着變化。

Convexity bias refers to the impact that the nonlinearity of a bond's price-yield curve has on the shape of the term structure. This impact is very small at the front end but can be quite significant at very long durations. A positively convex price-yield curve has the property that a given yield decline raises the bond price more than a yield increase of equal magnitude reduces it. All else equal, this property makes a high-convexity bond more valuable than a low-convexity bond, especially if the volatility is high. It follows that investors tend to accept a lower initial yield for a more convex bond because they have the prospect of enhancing their returns as a result of convexity. Because a long bond exhibits much greater convexity than a short bond, it can have a lower yield and yet, offer the same near-term expected return. Thus, in the absence of bond risk premia, the convexity bias would make the term structure inverted. In the presence of positive bond risk premia, the convexity bias tends to make the term structure humped —— because the negative effect of convexity bias overtakes the positive effect of bond risk premia only at long durations. An increase in the interest rate volatility makes the bias stronger and, thus, tends to make the term structure more humped.

凸度誤差是指債券的價格-收益率曲線的非線性對期限結構的影響。這種影響在短久期端很是小，但在長久期端可能會至關顯着。一個正凸度的價格-收益率曲線使得收益率降低帶來的債券價格提升程度大於收益率上升帶來的債券價格下降程度。其餘條件都相同的狀況下，這種特性使得高凸度債券比低凸度債券更有價值，特別是波動率高的狀況下。所以，對於凸度更大的債券，投資者傾向於接受較低的初始收益率，由於凸度有增長回報的效果。在近期預期回報相同的前提下，因爲長期債券比短時間債券具備更大的凸度，因此它的收益率可能會較低。所以，在沒有債券風險溢價的狀況下，凸度誤差將使期限結構倒掛。在存在正的債券風險溢價的狀況下，凸度誤差傾向於使期限結構隆起——因爲在長久期端凸度誤差的負面影響超過債券風險溢價的正面影響。收益率波動性的增長使得凸度誤差更大，所以趨向於使期限結構更加隆起。

The three determinants influence the shape of the term structure simultaneously, making it difficult to distinguish their individual effects. One central theme in this series has been that the shape of the term structure does not only reflect the market's rate expectations. Forward rates are good measures of the market's rate expectations only if the bond risk premia and the convexity bias can be ignored. This is hardly the case, even though a large portion of the short-term variation in the shape of the curve probably reflects the market's changing expectations about the future level and shape of the curve. The steepness of the curve on a given day depends mainly on the market's view regarding the rate direction, but in the long run, the impact of positive and negative rate expectations largely washes out. Therefore, the average upward slope of the yield curve is mainly attributable to positive bond risk premia. The curvature of the term structure may reflect all three components. On a given day, the spot rate curve is especially concave (humped) if market participants have strong expectations of future curve flattening or of high future volatility. In the long run, the reshaping expectations should wash out, and the average concave shape of the term structure reflects the concavity of the risk premium curve and the convexity bias.

三個決定因素同時影響期限結構的形狀，使得難以區分其各自的做用。這個系列的一箇中心主題是，期限結構的形狀不只僅反映了市場的收益率預期。只有債券風險溢價和凸度誤差能夠忽略，遠期收益率纔是市場收益率預期的良好指標。事實並不是如此，儘管很大程度上曲線形狀的短時間變化反映了市場對曲線將來水平和形狀的變化預期。曲線在某一天的陡峭程度主要取決於市場對收益率方向的見解，但從長遠來看，正負收益率預期的影響大部分被淘汰。所以，平均來看向上傾斜的收益率曲線主要歸咎於債券風險溢價。期限結構的曲率可能反映全部三個成分。在某一天，若是市場參與者對將來曲線變平或將來波動率變大有強烈的預期，則即期收益率曲線尤爲是上凸的。從長遠來看，曲線形變的預期會被淘汰，而平均來看期限結構的上凸形狀反映了風險溢價曲線的凸度和凸度誤差。

Decomposing Forward Rates Into Their Main Determinants

分解遠期收益率

Conceptually, each one-period forward rate can be decomposed to three parts: the impact of rate expectations; the bond risk premium; and the convexity bias. So far, this statement is just an assertion. In this subsection, we show intuitively why this relationship holds between the forward rates and their three determinants. We provide a more formal derivation in Appendix A (where we take into account the fact that the analysis is not instantaneous but yield changes occur over a discrete horizon, during which invested capital grows). In Appendix B, we tie some loose strings together by summarizing various statements about the forward rates and by clarifying the relations between these statements.

從概念上講，每一期遠期收益率均可以分解爲三個部分：收益率預期的影響、債券風險溢價和凸度誤差。到目前爲止，這個說法只是一個斷言。在本小節中，咱們直觀地展現爲何遠期收益率與其三個決定因素之間存在這種關係。咱們在附錄A中提供更正式的推導（在那裏咱們考慮到分析不是即時的，而是在收益率發生變化的一段時期）。在附錄B中，咱們經過總結關於遠期收益率的各類陳述以及理清這些陳述之間的關係，將一些鬆散的結論串聯在一塊兒。

Figure 1 shows how the yield change of an n-year zero-coupon bond over one period (dashed arrow) can be split to the rolldown yield change and the one-period change in an n-1 year constant-maturity spot rate \(s_{n-1}\) (\(\Delta s_{n-1} = s^*_{n-1} - s_{n-1}\)) (two solid arrows).³ A zero-coupon bond's price can be split in a similar way (see Appendix A). Thus, an n-year zero's holding-period return over the next period \(h_n\) is:

圖1顯示了在一個時期n年期零息債券的收益率變化（虛線箭頭）如何能夠分解爲滾動收益率的變化和n-1年期即期收益率\(s_{n-1}\)的變化（\(\Delta s_{n-1} = s^*_{n-1} - s_{n-1}\)，兩個實線箭頭）。零息債券的價格能夠以相似的方式分解（見附錄A）。所以，n年期零息債券的一（年）期持有期回報\(h_n\)爲：

\[ \begin{aligned} h_n &= \text{return if curve is unchanged} &+& [\text{return from the curve changes}] \\ &= \text{rolling yield} &+& [\text{percentage price change (at horizon)}] \\ &\approx \text{(one-period) forward rate} &+& [-\text{duration} * (\Delta s_{n-1}) + 0.5*\text{convexity} * (\Delta s_{n-1})^2] \end{aligned} \tag{1} \]

Figure 1 Splitting a Zero-Coupon Bond's One-Period Yield Change Into Two Parts

Equation (1) is based on the following relations. First, a bond's one-period horizon return given an unchanged yield curve is called the rolling yield. A zero-coupon bond's rolling yield equals the one-period forward rate (\(f_{n-1,n}\)). For example, if the four-year (five-year) constant-maturity rate remains unchanged at 9.5% (10%) over the next year, a five-year zero bought today at 10% can be sold next year at 9.5%, as a four-year zero; then the bonds horizon return is \(1.10^5 / 1.095^4 - 1 = 12.02\%\), which is the one-year forward rate between four- and five-year maturities (see Equation (12) in Appendix B). The second source of a zero's holding-period return, the price change caused by the yield curve shift, is approximated very well by duration and convexity effects for all but extremely large yield curve shifts.

等式（1）基於如下關係。首先，給定不變收益率曲線的債券一（年）期持有期回報稱爲滾動收益率。零息債券的滾動收益率等於一（年）期遠期收益率（\(f_{n-1,n}\)）。例如，若是四年（五年）收益率在將來一年保持不變，爲9.5%（10%），那麼今年以10%的收益率購買的五年期零息債券下一年將以9.5%的收益率出售，那麼債券持有期回報爲12.02%，即四年到五年期間的一年期遠期收益率（見附錄B中的等式（12））。零息債券回報的第二個來源，即收益率曲線變更引發的價格變更，能用久期和凸度很好的近似，除非收益率曲線發生很是大的變化。

It is more interesting to relate the forward rates to expected returns and expected rate changes than to the realized ones. We take expectations of both sides of Equation (1), split the bond's expected holding-period return into the short rate and the bond risk premium, and recall that \(E(\Delta s_{n-1})^2 \approx (Vol(\Delta s_{n-1}))^2\). Then we can rearrange the equation to express the one-period forward rate as a sum of the other terms:

相對於已實現收益率，將遠期收益率與預期回報和預期收益率變更相聯繫將會更有意義。咱們對等式（1）的兩邊取指望，將債券的預期持有期回報分爲短時間收益率和債券風險溢價兩部分，並回顧\(E(\Delta s_{n-1})^2 \approx (Vol(\Delta s_{n-1}))^2\)。而後，咱們能夠從新排列等式，將遠期收益率做爲其餘項的和：

\[ \text{Forward rate} \approx \text{short rate} + \text{duration} * E(\Delta s_{n-1}) + \text{bond risk premium} + \text{convexity bias}, \tag{2} \]

where bond risk premium = \(E(h_n - s_1)\) and convexity bias \(\approx -0.5 * \text{convexity}*(Vol(\Delta s_{n-1}))^2\).

其中債券風險溢價=\(E(h_n - s_1)\)，凸度誤差\(\approx -0.5 * \text{convexity}*(Vol(\Delta s_{n-1}))^2\)。

If we move the short rate to the left-hand side of the equation, we decompose the "forward-spot premium" (\(f_{n-1, n} - s_1\)) into a rate expectation term, a risk premium term and a convexity term (see Equation (11) in Appendix A). We interpret the expectations in Equation (2) as the market's rate and volatility expectations and as the expected risk premium that the market requires for holding long-term bonds. The market's expectations are weighted averages of individual market participants' expectations.

若是咱們將短時間收益率移動到等式的左邊，咱們將「遠期-即期溢價」（\(f_{n-1, n} - s_1\)）分解爲預期收益率、風險溢價和凸度項（見附錄A中的等式（11））。咱們將等式（2）中的預期值視爲市場收益率和波動率的預期值，以及持有長期債券所需的預期風險溢價。市場預期是個體市場參與者預期的加權平均值。

Some readers may wonder why our analysis deals with forward rates and not with the more familiar par and spot rates. The reason is the simplicity of the one-period forward rates. A one-period forward rate is the most basic unit in term structure analysis, the discount rate of one cash flow over one period. A spot rate is the average discount rate of one cash flow over many periods, whereas a par rate is the average discount rate of many cash flows —— those of a par bond —— over many periods. All the averaging makes the decomposition messier for the spot rates and the par rates than it is for the one-period forward rate in Equation (2). However, because the spot and the par rates are complex averages of the one-period forward rates, they too can be conceptually decomposed into the three main determinants.

一些讀者可能會想知道爲何咱們的分析涉及遠期收益率，而不是更熟悉的到期和即期收益率。緣由是一年期遠期收益率的簡單性。一年期遠期收益率是期限結構分析中最基本的單位，即一年期現金流貼現率。即期收益率是多期現金流的平均貼現率，而到期收益率是許多多期現金流的平均貼現率。全部的平均化使得對於即期收益率和到期收益率的分解與等式（2）中的一年期遠期收益率相比更困難。然而，因爲即期和到期收益率是一年期遠期收益率的複雜平均，它們也能夠在概念上分解爲三個主要決定因素。

Because the approximate decomposition in Equation (2) is derived mathematically without making specific economic assumptions, it is true in general. In reality, however, it is hard to make this decomposition because the components are not observable and because they vary over time. Further assumptions or proxies are needed for such a decomposition. In Figure 2, we use historical average returns to compute the bond risk premia and historical volatilities to compute the convexity bias —— together with the observable market forward rates (as of September 26, 1995) —— and back out the only unknown term in Equation (2): the expected spot rate change times duration. We also could divide this term by duration to infer the market's rate expectations. The rate expectations that we back out in Figure 2 suggest that the market expects small declines in short rates and small increases in long rates.

由於等式（2）中的近似分解是在數學意義上得出而沒有做出具體的經濟假設的，因此一般是正確的。然而，因爲各分項不可見而且隨時間而變化，因此實際上難以進行分解。這種分解須要進一步的假設或指代。在圖2中，咱們使用歷史平均回報來計算債券風險溢價，用歷史波動率來計算凸度誤差，以及可觀察的市場遠期收益率（截至1995年9月26日），推算出公式中惟一未知的項：預期即期收益率變更乘以久期。咱們也能夠除以久期以推斷市場的收益率預期。咱們在圖2中算出的預期收益率代表，市場預期短時間收益率小幅下跌，長期收益率小幅上漲。

Figure 2 Decomposing Forward Rates Into Their Components, Using Historical Average Risk Premia and Volatilities

If bond risk premia vary over time, the use of historical average risk premia may be misleading. As an alternative, we can use survey data or rate predictions based on a quantitative forecasting model to proxy for the market's rate expectations. In Figure 3, we use the consensus interest rate forecasts from the Blue Chip Financial Forecast to proxy for the market's rate expectations. In addition, we use implied volatilities from option prices to compute the convexity bias. These components can be used together with the one-year forward rates to back out estimates of the unobservable bond risk premia.

若是債券風險溢價隨時間而變化，使用歷史平均風險溢價可能會產生誤導。做爲替代，咱們可使用調查數據或基於量化預測模型的收益率預測來代替市場的收益率預期。在圖3中，咱們使用「Blue Chip金融預測」的一致收益率預測來代替市場的收益率預期。此外，咱們使用期權價格的隱含波動率來計算凸度誤差。這些組成部分能夠與一年期遠期收益率一塊兒使用，推算不可觀察的債券風險溢價的估計。

Figure 3 Decomposing Forward Hates Into Their Components, Using Survey Hate Expectations and Implied Volatilities

A comparison of Figures 2 and 3 shows that the two decompositions look similar up to the seven-year duration, but quite different beyond that point. The similarity of the convexity bias components in these two figures suggests that the use of historical or implied volatilities makes little difference, at least in this case. It is also clear that the Blue Chip survey predicted small declines in the short rates and small increases in the long rates, just as the inferred rate expectations in Figure 2. However, the predicted increases in long rates were smaller in this survey (where the largest increase was four basis points) than the inferred forecasts of Figure 2 (where the largest increase was eight basis points). Because the forward rate curve is the same in both figures, the smaller predicted rate increases lead to higher bond risk premia in Figure 3 than in Figure 2.⁴

圖2和圖3的比較顯示，兩個分解從七年久期開始看起來相似，可是七年久期以前差異比較大。這兩幅圖中的凸度誤差部分的類似性代表，使用歷史或隱含波動率幾乎沒有什麼區別，至少在這個例子的狀況下。很明顯，Blue Chip調查預測短時間收益率小幅下滑，長期收益率小幅上漲，正如圖2推測的收益率預期同樣。然而，本次調查中長期收益率的預測上升幅度（最大增幅爲4個基點）小於圖2的預測上升幅度（最大增幅爲8個基點）。因爲兩個圖中的遠期收益率曲線相同，因此較小的預測收益率上升致使圖3中的債券風險溢價高於圖2。

Figures 2 and 3 are snapshots of the forward rates and their components on one date. A comparison of similar decompositions over time would provide insights into the relative variability of each component. In Figure 4, we try to illustrate the impact of changing rate expectations and risk premia on the steepness of the US Treasury bill curve. The figure shows that on four recent dates the forwards always implied larger increases in the three-month rate than the market expected, based on surveys of bond market analysts. The difference is proportional to the required bond risk premium of longer bills over shorter bills (because bills exhibit negligible convexity, its impact can be ignored). Not surprisingly, this difference is always positive; moreover, it varies over time.

圖2和圖3是某一天的遠期收益率及其組成部分的截圖。隨着時間的推移，相似分解的比較將提供每一個組成部分的相對變化的看法。在圖4中，咱們試圖說明預期收益率變化和風險溢價對美國國庫券曲線陡峭程度的影響。該數字顯示，根據債券市場分析師的調查，近期四個交易日始終代表，三月期收益率相比於市場預期大幅上漲。差額與長期國庫券超出短時間國庫券的債券風險溢價成比例（由於國庫券的凸度影響能夠忽略）。絕不奇怪，這種差別老是正的；此外，它隨時間而變化。

Figure 4 Forward Implied Yield Changes versus Survey-Expected Yield Changes in the Treasury Bill Market, 1993-95

The time-variation in the bond risk premium in Figure 4 appears economically reasonable. In December 1993, after a long bull market, market participants were complacent, neither expecting much higher rates nor demanding much compensation for duration extension. After the Fed began to tighten the monetary policy, the market expected further rate increases. In addition, the increase in volatility and in risk aversion levels (perhaps caused by own losses as well as the well-publicized losses of other investors) increased the required bond risk premia. By the end of 1994, the market was extremely bearish, expecting almost 100 basis points higher three-month rates in six months. However, the forwards were implying almost 200 basis points higher rates —— the difference reflects an abnormally large risk premium. In 1995, the bond market rallied strongly as the market's expectations for further Fed tightening receded and turned into expectations of easing policy. However, a large part of this rally was caused by the collapse in the required bond risk premium, perhaps reflecting lower inflation uncertainty and higher wealth that reduced the market's risk perceptions and risk aversion. In general, the time-variation in required returns appears to have contributed as much to the changing slope of the yield curve as has the time-variation in the market's interest rate expectations.⁵ Finally, we note that the time-variation in the estimated bond risk premia has been very market-directional over the past two years; this may not always be the case.

圖4中債券風險溢價的時變性彷佛在經濟上是合理的。1993年12月，通過長時間的牛市，市場參與者很是知足，既不指望更高的收益率，也不要求增長久期。美聯儲開始收緊貨幣政策後，市場預期收益率進一步上調。此外，波動率和風險規避水平的增長（可能由自身的損失以及其餘投資者的普遍的損失引發）提升了債券風險溢價。到1994年末，市場十分悲觀，在六個月內預期三月期收益率漲幅將近100個基點。然而，遠期收益率隱含漲幅高於近200個基點，差別反映出風險溢價異常大。1995年，債券市場強勁反彈，市場對美聯儲進一步緊縮政策的預期降低，並轉爲預期放鬆政策。然而，這個反彈的很大一部分是因爲債券風險溢價的崩潰所致，也許是反映了更低的通脹不肯定性和更高的財富，從而下降了市場的風險認知和風險規避。通常來講，所需回報的時變性彷佛對收益率曲線的斜率變化有很大的做用，這與市場收益率預期的時變性同樣。最後，咱們注意到，債券風險預期的時變性在過去兩年中表現出很強的市場方向性；這可能並不是老是如此。

DECOMPOSING EXPECTED RETURNS OF BOND POSITIONS

分解債券頭寸的預期回報

Five Alternative Expected Return Measures

五種預期回報度量方法

Our framework for decomposing the yield curve also provides a framework for systematic relative value analysis of government bonds with known cash flows. We can evaluate all bond positions' expected returns comprehensively, yet with simple and intuitive building blocks. We emphasize that relative value analysis should be based on near-term expected return differentials, not on yield spreads, which are only one part of them. That is, total return investors should care more about expected returns than about yields. Thus, our approach brings fixed-income investors closer to mean-variance analysis in which various positions are evaluated based on the trade-off between their expected return and return volatility.

咱們分解收益率曲線的框架也爲已知現金流的政府債券的相對價值分析提供了系統性框架。咱們能夠用簡單直觀的模塊全面評估全部債券頭寸的預期回報。咱們強調，相對價值分析應以近期預期回報差別爲基礎，而不是利差，利差僅做爲近期預期回報差的一部分。也就是說，總回報投資者應該更多關注預期回報而不是收益率。所以，咱們的方法使固定收益投資者更熟悉均值-方差分析，根據預期回報和回報波動率之間的權衡來評估各類頭寸。

Equation (1) shows that a zero's holding-period return is a sum of its return given an unchanged yield curve and its return caused by the changes in the yield curve. The return given an unchanged yield curve is called the rolling yield because it is a sum of the zero's yield and the rolldown return. The return caused by changes in the yield curve can be approximated well by duration and convexity effects. Taking expectations of Equation (1) and splitting the rolling yield into yield income and rolldown return, the near-term expected return of a zero is:

等式（1）代表零息債券持有期回報是給定不變收益率曲線狀況下的回報及收益率曲線變化引發的回報之和。給定不變收益率曲線狀況下的回報被稱爲滾動收益率，由於它是零息債券收益率和下滑回報之和。由收益率曲線變化引發的回報能夠經過久期和凸度很好地近似。對等式（1）求指望，將滾動收益率分解爲收益率回報和下滑回報，零息債券近期預期回報爲：

\[ \begin{aligned} \text{Expected return} &\approx \text{Yield income} \\ &+ \text{Rolldown return} \\ &+ \text{Value of convexity} \\ &+ \text{Expected capital gain/loss from the rate "view"} \end{aligned} \tag{3} \]

For details, see Equation (9) in Appendix A or the notes below Figure 5. A similar relation holds approximately for coupon bonds, and we will describe the three-month expected return of some on-the-run Treasury bonds as the sum of the four components in the right-hand side of Equation (3).⁶

詳情請參見附錄A中的等式（9）或圖5下面的附註。相似的關係對付息債券近似成立，咱們將把一些活躍國債的三個月預期回報分解成等式（3）右側的四個份量。

Figure 5 Three-Month Expected Return Measures and Their Components, as of 26 Sep 95

This framework is especially useful when evaluating positions of two or more government bonds, such as duration-neutral barbells versus bullets. We first compute expected return separately for each component and then compute the portfolio's expected return by taking a market-value weighted average of all the components' expected returns.

當評估兩個或多個政府債券的頭寸時，這種框架特別有用，例如久期中性的槓鈴組合與子彈組合。咱們首先針對每一個份量分別計算預期回報，而後經過對全部份量的預期回報進行市值加權平均來計算投資組合的預期回報。

It may be helpful to show step by step how the expected return measures are improved, starting from simple yields and moving toward more comprehensive measures.

經過逐步展現如何從簡單的收益率到更全面的方法來改進預期回報的度量，可能會對讀者有所幫助。

• A bond's yield income includes coupon income, accrued interest and the accretion/amortization of price toward par value. Yield to maturity is the correct return measure if all interim cash flows can be reinvested at the yield and the bond can be sold at its purchasing yield.⁷ Yield ignores the rolldown return the bond earns if the yield curve stays unchanged.

• 債券的收益率回報包括息票收入、應計利息和價格對面值的增長/攤銷。若是全部中期現金流能夠以到期收益率再投資，債券能夠以其購買時的收益率出售，則到期收益率是正確的回報率。若是到期收益率曲線保持不變，收益率將忽略下滑回報。

• Rolling yield is a better expected return proxy if an unchanged curve is a reasonable base case. Yet, it ignores the value of convexity and, thus, implicitly assumes no rate uncertainty. Thus, the rolling yield measures expected return if no curve change and no volatility is expected.

• 若是曲線不變，滾動收益率能更好地表明預期回報。然而，它忽略了凸度價值，所以隱含地假定收益率不存在不肯定性。所以，若是沒有曲線變化而且沒有預期波動，則滾動收益率能夠度量預期回報。

• Combining the rolling yield with the value of convexity improves the expected return measure further. In Part 5 of this series, we showed that a bond's convexity-adjusted expected return equals the sum of the rolling yield and the value of convexity. This measure recognizes the impact of rate uncertainty but implies that no change is expected in the yield curve. Empirical evidence described in Part 2 suggests that an unchanged yield curve is often a reasonable base "view".

• 將滾動收益率與凸度價值結合，進一步優化對預期回報的度量。在本系列的第5部分中，咱們代表，債券的凸度調整預期回報等於滾動收益率與凸度價值的總和。這一度量方法認識到收益率不肯定性的影響，但同時預期收益率曲線不會發生變化。第2部分的實證證據代表，收益率曲線不變一般是一個合理的基本「觀點」。

• If investors want, they can replace the prediction of an unchanged curve with some other rate (or spread) "view". One possibility is to use survey-based information of the market's current rate forecasts; such an approach may be useful for backing out the market's required return for each bond. Alternatively, investors may ignore the market view and input either their own rate views or an economist's subjective rate forecasts or rate predictions from some quantitative model. For example, the predictors identified in Part 4 of this series can be used to forecast the changes in the long rates. The impact of any rate view is approximated by the expected yield change sealed by duration (see Equation (10) in Appendix A), which may be added to the convexity-adjusted expected return. The sum gives us the "expected return with a view" —— the four-term expected return measure in Equation (3). However, this equation is a perfect description of expected returns only for bonds that lie on the fitted curve. Thus, the relative value measures above ignore "local" or bond-specific richness or cheapness relative to the curve.

• 若是投資者想要，他們能夠用一些其餘收益率（或利差）觀點替代不變曲線的預測。一種可能的方法是使用基於調查的市場收益率預測的信息，這種方法可能有助於推算出市場對每一個債券所要求的回報。或者，投資者能夠忽視市場觀點，並從一些經濟學家的主觀預測或量化模型的預測中獲得觀點。例如，本系列第4部分中肯定的預測因子可用於預測長期收益率的變化。任何收益率觀點的影響都是由久期結合預期收益率變化近似（見附錄A中的等式（10）），能夠將其加到凸度調整後的預期回報中。這個總和給了咱們「帶觀點的預期回報」——等式（3）中的四項預期回報度量。然而，這個等式是對落在擬合曲線上的債券預期回報的完美描述。所以，上述相對價值度量忽略了債券相對於曲線「局部的」或債券特定的估值誤差。

• Many technical factors can make a specific bond "locally" rich or cheap (relative to adjacent-maturity bonds), or they can make a whole maturity sector rich or cheap relative to the fitted curve. Such factors include supply effects (temporary price pressure on a sector caused by new issuance), demand effects (maturity limitations or preferences of important market participants —— for example, the richness of quarter-end bills), liquidity effects (lower transaction costs for on-the-runs versus off-the-run, for 30-year bonds versus 25-year bonds, for Treasury bills versus duration-matched coupon bonds, etc.), coupon effects (motivated by tax benefits, accounting rules, etc.), and above all, the financing effects (the "special" repo income that is common for on-the-runs).⁸ Fortunately, it is easy to add to the four-term expected return measures the financing advantage and two local cheapness measures —— the spread off the fitted curve and the expected cheapening toward the fitted curve. The five-term expected return measures are comprehensive measures of total expected returns —— ignoring small approximation errors, they incorporate all sources of expected return for noncallable government bonds.⁹

• 許多技術因素可使特定債券局部地高估或低估（相對於相鄰到期債券），或者使整個期限區間相對於擬合曲線高估或低估。這些因素包括供給效應（由新發行形成的暫時的價格壓力）、需求效應（期限限制或重要市場參與者的偏好，例如季末國庫券的被高估）、流動性效應（例如活躍券對不活躍券、30年期債券對25年期債券、國庫券對久期匹配的付息債券而言具備較低的交易成本）、票息效應（由稅收優惠、會計規則等引起）、最重要的是融資效應（對活躍券而言常見的「特殊」回購回報）。幸運的是，很容易將融資優點和兩個局部的低估度量（與擬合曲線的利差，相對於擬合曲線的預期低估量）添加到四因子預期回報度量中。五因子預期回報度量是預期回報總量的綜合度量指標（忽略了小的近似偏差），它們包含了非可贖回的政府債券全部預期回報來源。

As a numerical illustration, Figure 5 shows the various expected return measures for three bonds (the three-month Treasury bill and the three-year and ten-year on-the-run Treasury notes) and for the barbell combination of the three-month bill and the ten-year bond. In this example, we use as much market-based data as possible: for example, implied volatilities, not historical, to estimate the value of convexity, and the "view" (rate predictions) based on survey evidence of the market's rate expectations, not on a quantitative forecasting model. All the numbers are based on the market prices as of September 26, 1995.

做爲數值例子，圖5顯示了三種債券（三月期國庫券、三年期和十年期國債）以及三月期國庫券-十年期債券槓鈴組合的各類預期回報度量。在這個例子中，咱們使用盡量多的基於市場的數據：例如，用隱含波動率而不是歷史波動率來估計凸度價值，以及基於市場收益率預期的調查證據而不是量化預測模型造成的收益率「觀點」（收益率預測）。全部數字都是根據1995年9月26日的市場價格計算的。

The top panel of Figure 5 shows how nicely the different components of expected returns can be added to each other. Moreover, the barbell's expected return measures are simply the market-value weighted averages of its components' expected returns. In this case, the yield income, the rolldown return and the value of convexity are all higher for the longer bonds. In contrast, the duration impact of the market's rate view is negative, because the Blue Chip survey suggested that the market expected small increases in the three-year and the ten-year rates over the next quarter. The local rich/cheap effect is negative for the shorter instruments but positive for the ten-year note; the reason is that the negative yield spread and the expected cheapening of the ten-year note are not sufficient to offset the ten-year note's high repo market advantage. According to all five expected return measures, the barbell has a lower expected return than the duration-matched bullet.

圖5的上半部分顯示了預期回報的不一樣組成部分能夠如何更好地相互添加。此外，槓鈴組合的預期回報度量只是其組成部分預期回報的市值加權平均。在這種狀況下，長期債券的收益率回報、下滑回報和凸度價值均較高。相比之下，市場收益率觀點的久期影響是負的，由於Blue Chip調查顯示，市場預期下個季度的三年期和十年期收益率將小幅上漲。局部高估/低估效應對於期限較短的國庫券爲負數，但對十年期債券爲正，緣由是負的利差和預期十年期債券被低估的程度不足以抵消十年期債券在回購市場的優點。根據全部五項預期回報度量，槓鈴組合的預期回報比久期匹配子彈組合低。

Figure 6 shows the five different expected return curves plotted on the three bonds' durations. In this case, the simplest expected return measure (yield income) and the most comprehensive measure (total expected return) happen to be quite similar. In general, the relative importance of the five components may be dramatically different from that in Figure 5 where the yield income dominates. The longer the asset's duration and the shorter the investment horizon, the greater is the relative importance of the duration impact. It is worth noting that realized returns can be decomposed in the same way as the expected returns and that the duration impact typically dominates the realized returns even more.¹⁰

圖6顯示了在三個債券關於久期繪製的五種不一樣的預期回報曲線。在這種狀況下，最簡單的預期回報度量（收益率回報）和最全面的度量（總預期回報）偏偏至關。通常來講，五個組成部分的相對重要性可能與圖5中收益率回報占主導地位的狀況有顯着差別。資產久期越長、投資期限越短，久期影響的相對重要性就越大。值得注意的是，實現的回報能夠以與預期回報相同的方式分解，久期的影響一般更多主導實現的回報。

Figure 6 Expected Returns at a Three-Month Bill, a Three-Year Bond and a Ten-Year Bond, as of 26 Sep 95

The total expected returns, if estimated carefully, should produce the most useful signals for relative value analysis because they include all sources of expected returns. Yield spreads may be useful signals, but they are only a part of the picture. Therefore, we advocate the monitoring of broader expected return measures relative to their history as cheapness indicators —— just as yield spreads are often monitored relative to their history.

若是仔細估計，總預期回報應該產生最有用的相對價值分析信號，由於它們包括全部預期回報來源。利差多是有用的信號，但它們只是一部分。所以，咱們主張監測預期回報度量相對於其歷史水平的差別做爲低估指標，正如一般監測利差相對於其歷史水平的差別同樣。

The components of expected returns discussed above are not new. However, few investors have combined these components into an integrated framework and based their historical analysis on broad expected return measures. An additional useful feature of this framework is that all types of government bond trades can be evaluated consistently within it: the portfolio duration decision (market-directional view); the maturity-sector positioning and barbell-bullet decision (curve-reshaping view); and the individual issue selection (local cheapness view). With small modifications, the framework can be extended to include the cross-country analysis of currency-hedged government bond positions. Other possible future extensions include the analysis of foreign exchange exposure and the analysis of spread positions between government bonds and other fixed-income assets.

上述關於預期回報的組成部分的討論並不新鮮。然而，不多有投資者將這些組成部分組合成一個綜合框架，並將他們的歷史分析基於普遍的預期回報度量之上。這個框架的一個額外的功能是全部類型的政府債券交易能夠在其中一致地評估：投資組合久期的制定（市場的方向性觀點）；期限配比和槓鈴-子彈組合的制定（曲線形變觀點）和個券選擇的制定（局部低估觀點）。通過少許修改，能夠將框架擴大到包括貨幣對衝的跨國政府債券頭寸的分析。其餘可能的將來擴展包括外匯敞口分析以及政府債券與其餘固定收益資產之間的利差分析。

We note some reservations. Even if two investors use the same general framework and the same type of expected return measure, they may come up with different numbers because of different data sources and different estimation techniques.

咱們注意到一些遺留問題。即便兩個投資者使用相同的通常框架和相同的預期回報度量，也可能會由於不一樣的數據來源和不一樣的估算技術而產生不一樣的結果。

• The whole analysis can be made with any raw material; we emphasize the importance of good-quality inputs. Various candidates for the raw material include on-the-run and off-the-run government bonds, STRIPS, Eurodeposits, swaps, and Eurodeposit futures. This multitude of course opens the possibility of trading between these curves if we can assess how various characteristics (say, convexity) are priced in each curve. The most common approach is first to estimate the spot curve (or discount function) using a broad universe of coupon government bonds as the raw material, and then to compute the forward rates and other relevant numbers. In European bond markets, the liquid swap curve (using cash Eurodeposits and swaps as the raw material) has gained more of a benchmark status. Of course, some credit and tax-related spread may exist between the swap curve and the government bond yield curve. Recently, yet another approach has become popular: Eurodeposit futures prices are used as the raw material. In this case, the forward rates are computed by adjusting for the convexity difference between a futures contract and a forward contract, and only then are spot rates computed from the forwards.

• 整個分析能夠用任何原始資料進行，咱們強調輸入優質數據的重要性。原始資料包括活躍和不活躍的政府債券、零息債券（STRIPS）、歐洲存款、互換和歐洲存款期貨。固然，若是咱們可以評估每一個曲線中的各類特性（如凸度）是如何訂價的，那麼很大程度上能夠提供這些曲線之間交易的可能性。最多見的作法是首先用普遍的付息政府債券做爲原始資料估算即期收益率曲線（或貼現函數），而後計算遠期收益率和其餘相關數字。在歐洲債券市場，流動性好的互換曲線（以現金歐洲存款和互換爲原始資料）已經更多地最爲基準。固然，互換曲線和政府債券收益率曲線之間可能存在一些信用和稅收相關的利差。最近，另外一種方法已經變得流行起來：以歐元存貨期貨價格爲原始資料。在這種狀況下，遠期收益率是經過調整期貨合約與遠期合約之間的凸度差來計算的，而後纔是從遠期收益率計算的即期收益率。

• Some components of expected returns are easier to measure —— and less debatable —— than others. The yield income is relatively unambiguous. The rolldown return and the local rich/cheap effects depend on the curve-fitting technique. The value of convexity depends on the volatility input and, thus, on the volatility estimation technique. The rate "view," the fourth term, can be based on various approaches, such as quantitative modeling or subjective forecasting that relies on fundamental or technical analysis. Even the quantitative approach is not purely objective because infinitely many alternative forecasting models and estimation techniques exist. Forecasting rate changes is of course the most difficult task as well as the one with greatest potential rewards and risks. Forecasting changes in yield spreads may be almost as difficult. The short-term returns of most bond positions depend primarily on the duration impact (rate changes or spread changes). However, even if investors cannot predict rate changes, they may earn superior returns in the long run —— and with less volatility —— by systematically exploiting the more stable sources of expected return differentials across bonds: yields; rolldown returns; value of convexity; and local rich/cheap effects. More generally, while the total expected return differentials are, in theory, better relative value indicators than the yield spreads, in practice, measurement errors conceivably can make them so noisy that they give worse signals. Therefore, it is important to check with historical data that any supposedly superior relative value tools would have enhanced the investment performance at least in the past.

• 預期回報的一些組成部分比其餘部分更容易度量。收益率回報相對明顯。下滑回報和局部高估/低估效應取決於曲線擬合技術。凸度價值取決於波動率，所以取決於波動率估計技術。第四項——收益率「觀點」，能夠基於各類方法獲得，例如依賴於基本面或技術分析的量化模型和主觀預測。即便是量化方法也不是純粹客觀的，由於存在無數多個替代預測模型和估計技術。預測收益率變化固然是最困難的任務，也是最有潛在回報和風險的。預測利差變化可能幾乎一樣困難。大多數債券頭寸的短時間回報主要取決於久期的影響（收益率或利差變化）。然而，即便投資者沒法預測收益率變更，長期而言也可能得到較高的回報和較小的波動率——經過系統地挖掘更穩定的，不一樣債券的預期回報差別的來源：收益率回報、下滑回報、凸度價值和局部高估/低估的影響。更通常地說，雖然總預期回報差別在理論上是比利差更好的相對價值指標，但在實踐中，測量偏差可使它們的噪聲很大，進而產生更差的交易信號。所以，重要的是要用歷史數據來檢驗，任何所謂優越的相對價值工具，至少在過去都能提升投資業績。

Link to Scenario Analysis

與情景分析相聯繫

Many active investors base their investment decisions on subjective yield curve views, often with the help of scenario analysis. Our framework for relative value analysis is closely related to scenario analysis. It may be worthwhile to explore the linkages further.

許多積極的投資者藉助情景分析的幫助，將他們的投資決策創建在收益率曲線的主觀觀點上。咱們的相對價值分析框架與情景分析密切相關。可能值得進一步探討聯繫。

An investor can perform the scenario analysis of government bonds in two steps. First, the investor specifies a few yield curve scenarios for a given horizon and computes the total return of his bond portfolio —— or perhaps just a particular trade —— under each scenario. Second, the investor assigns subjective probabilities to the different scenarios and computes the probability-weighted expected return for his portfolio. Sometimes the second step is not completed and investors only examine qualitatively the portfolio performance under each scenario. However, we advocate performing this step because investors can gain valuable insights from it. Specifically, the probability-weighted expected return is the "bottom-line" number a total return manager should care about. By assigning probabilities to scenarios, investors also can explicitly back out their implied views about the yield curve reshaping and about yield volatilities and correlations.

投資者能夠分兩步進行政府債券的情景分析。首先，投資者在給定的期限內指定了一些收益率曲線情景，並計算每一個情景下債券投資組合的總回報，或者也可能只是特定的交易。第二，投資者將主觀機率分配給不一樣的情景，並計算其投資組合的機率加權預期回報。有時候第二步不須要完成，投資者只會在每一個情景下定性地檢查投資組合的表現。可是，咱們主張執行這一步，由於投資者能夠從中得到寶貴的看法。具體來講，機率加權的預期回報是總回報經理應該關心的「底線」數字。經過將機率分配給情景，投資者也能夠明確地推算出其關於收益率曲線形變以及關於收益率波動率和相關係數的隱含觀點。

In scenario analysis, investors define the mean yield curve view and the volatility view implicitly by choosing a set of scenarios and by assigning them probabilities. In contrast, our framework for relative value analysis involves explicitly specifying one yield curve view (which corresponds to the probability-weighted mean yield curve scenario) and a volatility view (which corresponds to the dispersion of the yield curve scenarios). Either way, the yield curve view determines the duration impact and the volatility view determines the value of convexity (and these views together approximately define the expected yield distribution).

在情景分析中，投資者經過選擇一組情景並分配其機率來隱含地定義平均曲線觀點和波動率觀點。相比之下，咱們的相對價值分析框架包括明確指定一個收益率曲線觀點（對應於機率加權的平均曲線情景）和波動率觀點（其對應於收益率曲線情景的分散程度）。不管哪一種方式，收益率曲線觀點決定久期影響，波動率觀點肯定凸度價值（這些觀點一塊兒大概界定了預期收益率分佈）。

Figure 7 presents a portfolio that consists of five equally weighted zero-coupon bonds with maturities of one to five years and (annually compounded) yields between 6% and 7%. The portfolio's maturity, and its Macaulay duration —— initially is three years. Over a one-year horizon, each zero's maturity shortens by one year. We specify five alternative yield curve scenarios over the horizon: parallel shifts of +100 basis points and -100 basis points; no change; a yield increase combined with a curve flattening; and a yield decline combined with a curve steepening (see Figure 8). We compute the one-year holding-period returns for each asset and for the portfolio under each scenario. In particular, the neutral scenario shows the rolling yield that each zero earns if the yield curve remains unchanged. We can evaluate each scenario separately. However, such analysis gives us limited insight —— for example, the last column in Figure 7 shows just that bearish scenarios produce lower portfolio returns than bullish scenarios.

圖7顯示了一個投資組合，其中包括五個等權的零息債券，期限爲一至五年，收益率（每一年複利）在6%至7%之間。投資組合的期限及其Macaulay久期最初爲三年。在一年的期限內，每一個零息債券的期限將縮短一年。咱們在持有期內指定了五種可能的收益率曲線情景：平移+100個基點和-100個基點、不變、收益率增長與曲線變平相結合、收益率降低與曲線變陡相結合（見圖8）。咱們計算每種情景下每種資產和投資組合的一年持有期回報。特別地，若是收益率曲線保持不變，則中性情景顯示每一個零息債券的滾動收益率。咱們能夠分別評估每一個情景。然而，這樣的分析給咱們提供了有限的洞察力——僅僅是熊市情景產生的投資組合回報比牛市情景更低，正如圖7中的最後一列欄顯示的。

Figure 7 Scenario Analysis and Expected Band Returns

Figure 8 Various Yield Curve Scenarios

In contrast, if we assign probabilities to the scenarios, we can back out many numbers of potential interest. We begin with a simple example in which we only use the two first scenarios, parallel shifts of 100 basis points up or down. If we assign these scenarios equal probabilities (0.5), the expected return of the portfolio is 7.04% (\(=0.5*5.02 + 0.5*9.06\)). On average, these scenarios have no view about curve changes; yet, this expected return is four basis points higher than the expected portfolio return given no change in the curve (that is, the 7% rolling yield computed in the neutral scenario). This difference reflects the value of convexity. If we only use one scenario, we implicitly assume zero volatility, which leads to downward-biased expected return estimates for positively convex bond positions. If we use the two first scenarios (bear and bull), we implicitly assume a 100-basis-point yield volatility; this assumption may or may not be reasonable, but it certainly is more reasonable than an assumption of no volatility. This example highlights the importance of using multiple scenarios to recognize the value of convexity. (The value is small here, however, because we focus on short-duration assets that have little convexity.)

相比之下，若是咱們將機率分配給情景，咱們能夠推算出許多潛在的有趣觀點。咱們從一個簡單的例子開始，只使用前兩個情景：上下平移100個基點。若是咱們分配這些情景相等機率（0.5），投資組合的預期回報爲7.04%（\(=0.5*5.02 + 0.5*9.06\)）。平均而言，這些情景認爲曲線沒有變化。然而這一預期回報高於曲線沒有變化情境下的預期回報四個基點（即在中性情景中計算出的7%滾動收益率）。這個差別反映了凸度價值。若是咱們只使用一種情景，咱們隱含地假定零波動率，這致使了正凸度債券頭寸的預期回報的被低估。若是咱們使用前兩個情景（熊市和牛市），咱們隱含地假設100個基點的收益率波動率，這個假設可能合理也可能不合理，可是確定比沒有波動的假設更合理。此例強調了使用多個情景來識別凸度價值的重要性。（這裏的凸度價值很小，是由於咱們專一於幾乎沒有凸度的短久期資產。）

Now we return to the example with all five yield curve scenarios in Figure 8. As an illustration, we assign each scenario the same probability (\(p_i = 0.2\)). Then, it is easy to compute the portfolio's probability-weighted expected return:

如今咱們回到圖8中全部五個收益率曲線情景的例子。做爲一個例證，咱們爲每一個情景分配相同的機率（\(p_i = 0.2\)）。那麼，很容易計算投資組合的機率加權預期回報：

\[ E(h_p) = \sum_{i=1}^5 p_i * h_i = 0.2*(5.02+9.06+7.00+5.51+7.51) = 6.82 \tag{4} \]

Given these probabilities, we can compute the expected return for each asset, and it is possible to back out the implied yield curve views. The lower panel in Figure 7 shows that the mean yield change across scenarios is +10 basis points for each rate (because the bear-flattener and the bull-steepener scenarios are not quite symmetric in magnitude in this example), implying a mild bearish bias but no implied curve-steepness views. In addition, we can back out the implied basis-point yield volatilities (or return volatilities) by measuring how much the yield change (or return) outcomes in each scenario deviate from the mean. These yield volatility levels are important determinants of the value of convexity. The last line in Figure 7 shows that the volatilities range from 80 to 66 basis points, implying an inverted term structure of volatility. Finally, we can compute implied correlations between various-maturity yield changes; the curve behavior across the five scenarios is so similar that all correlations are 0.92 or higher (not shown). Note that all correlations would equal 1.00 if only the first three scenarios were used; the imperfect correlations arise from the bear-flattener and the bull-steepener scenarios.

考慮到這些機率，咱們能夠計算每一個資產的預期回報，而且能夠推算出隱含的收益率曲線觀點。圖7中的下半部分顯示，不一樣情景的平均收益率變化爲+10個基點（由於在這個例子中，熊平和牛陡的情景在大小上不是很對稱），這意味着溫和的看跌誤差但沒有隱含曲線變陡的觀點。此外，咱們能夠經過測量每種狀況下的收益率變化（或回報）結果與平均值的誤差多少來推算隱含的基點收益率波動率（或回報波動率）。這些收益率波動率水平是凸度價值的重要決定因素。圖7的最後一行顯示，波動率幅度在80到66個基點之間，這意味着波動率期限結構的倒掛。最後，咱們能夠計算不一樣期限收益率變化之間的隱含相關性，五個情景中的曲線行爲很是類似，全部相關性均爲0.92或更高（未顯示）。注意，若是僅使用前三種狀況，全部相關性將等於1.00；這種不徹底的相關性來自於熊平和牛陡情景。

Whenever an investor uses scenario analysis, he should back out these implicit curve views, volatilities and correlations —— and check that any biases are reasonable and consistent with his own views. Without assigning the probabilities to each scenario, this step cannot be completed; then, the investor may overlook hidden biases in his analysis, such as a biased curve view or a very high or low implicit volatility assumption which makes positive convexity positions appear too good or too bad. If investors use quantitative tools —— such as scenario analysis, mean-variance optimization, or the approach outlined in this report —— to evaluate expected returns, they should recognize the importance of their rate views in this process. Strong subjective views can make any particular position appear attractive. Therefore, investors should have the discipline and the ability to be fully aware of the views that are input into the quantitative tool.

每當投資者使用情景分析時，他應該推算出這些隱含的曲線觀點、波動率和相關性，並檢查任何誤差是否合理和符合本身的觀點。沒有爲每一個情景分配機率，此步驟沒法完成；那麼投資者在分析中可能會忽視隱藏的誤差，好比有誤差的曲線觀點，或是很是高或低的隱含波動率假設，這使得正凸度的頭寸看起來太好或太差。若是投資者使用量化工具，例如情景分析，均值-方差優化或本報告中概述的方法來評估預期回報，那麼他們應該在這個過程當中認識到他們的收益率觀點的重要性。強烈的主觀觀點可使任何特定頭寸顯得有吸引力。所以，投資者應該具備紀律和能力充分認識歸入量化工具的（收益率）觀點。

In addition to the implied curve views, we can back out the four components of expected returns discussed above. In this example, we only analyze bonds that lie 「on the curve" and thus can ignore the fifth component, the local rich/cheap effects. (1) We measure the yield income from the portfolio by a market-value weighted average yield of the five zeros, which is 6.50% (see footnote 7). (2) Each asset's rolldown return is the difference between the horizon return given an unchanged yield curve and the yield income. Figure 7 shows that the horizon return for the portfolio is 7% in the neutral scenario; thus, the portfolio's (market-value weighted average) rolldown return is 50 basis points (= 7% - 6.5%). Note that the rolldown return is larger for longer bonds, reflecting the fact that the same rolldown yield change (25 basis points) produces larger capital gains for longer bonds. (3) The value of convexity for each zero can be approximated by \(0.5*\text{convexity at horizon}*(\text{basis-point yield volatility})^2*(1 + \text{rolling yield}/100)\). Using the implicit yield volatilities in Figure 7, this value varies between 0.6 and 4.5 basis points across bonds. The portfolio's value of convexity is a market-value weighted average of the bond-specific values of convexity, or roughly two basis points. (4) The duration impact of the rate "view" for each bond equals \((-\text{duration at horizon})*(\text{expected yield change})*(1 + \text{rolling yield}/100)\). The last term is needed because each invested dollar grows to \((1 + \text{rolling yield}/100)\) by the end of horizon when the repricing occurs. The core of the duration impact is the product of duration and expected yield change. The expected yield change refers to the change (over the investment horizon) in a constant-maturity rate of the bonds horizon maturity. In Figure 7, all rates are expected to increase by ten basis points, and the duration impact on specific bonds' returns varies between 0 and -40 basis points. The portfolio's duration impact is a market-value weighted average of bond-specific duration impacts, or about -20 basis points.¹¹

• 除了隱含的曲線觀點外，咱們還能夠推算上面討論的預期回報的四個組成部分。在這個例子中，咱們只分析「曲線上的」債券，從而忽略了第五個組成部分，局部的高估/低估的影響。（1）咱們經過五個零息債券的市值加權來衡量投資組合的收益率回報，即6.50%（見注7）；（2）每一個資產的下滑回報是一個不變收益率曲線狀況下的持有期回報與收益率回報之間的差，圖7顯示投資組合的持有期回報爲7%。所以，投資組合（市值加權平均）下滑回報爲50個基點（= 7% - 6.5%）。請注意，長期債券的下滑回報較大，反映出一樣的收益率下滑變化（25個基點）對長期債券產生較大的資本回報；（3）每一個零息債券的凸度價值能夠近似爲\(0.5*\text{期末凸度}*(\text{基點收益率波動率})^2*(1 + \text{滾動收益率}/100)\)；使用圖7中的隱含收益率波動率，該值變化區間在0.6和4.5基點。投資組合的凸度價值是債券凸度價值的市值加權平均值，大體兩個基點；（4）每一個債券收益率「觀點」的久期影響等於\((-\text{期末久期})*(\text{預期收益率變化})*(1 + \text{滾動收益率}/100)\)。最後一項是必要的，由於當從新訂價發生時，每一個投資美圓在持有期末增加到\((1 + \text{滾動收益率}/100)\)。久期影響的核心是久期和預期收益率變化的乘積。預期收益率變更是指固按期限上的收益率變化（在投資期限內）。在圖7中，全部收益率預期將增長10個基點，久期對特定債券回報的影響在0到-40個基點之間變化。投資組合的久期影響是特定債券久期影響的市值加權平均，約爲-20個基點。

Figure 9 shows that the four components add up to the total probability-weighted expected return of 6.82%. Decomposing expected returns into these components should help investors to better understand their own investment positions. For example, they can see what part of the expected return reflects static market conditions and what part reflects their subjective market view. Unless they are extremely confident about their market view, they can emphasize the part of expected return advantage that reflects static market conditions. In our example, the duration effect is small because the implied rate view is quite mild (ten basis points) and the one-year horizon is relatively long (the "slower" effects have time to accrue). With a shorter horizon and stronger rate views, the duration impact would easily dominate the other effects.

圖9顯示，這四個組成部分加起來的總機率加權預期回報爲6.82%。將預期回報分解爲這些組成部分應有助於投資者更好地瞭解本身的頭寸。例如，他們能夠看到預期回報的哪一部分反映了靜態市場情況，哪些部分反映了他們的主觀市場觀點。除非他們對市場觀點很是有信心，不然他們能夠強調反映靜態市場條件的預期回報優點部分。在咱們的示例中，久期效應很小，由於隱含的收益率觀點至關溫和（十個基點），一年的投資期限相對較長（「較慢」的影響有時間去逐漸累積）。若是投資期限更短，收益率觀點更強，久期的影響將很容易超過其餘的影響。

Figure 9 Decomposing the Total Expected Return into Four Components

APPENDIX A. DECOMPOSING THE FORWARD RATE STRUCTURE INTO ITS MAIN DETERMINANTS

附錄A：將遠期收益率結構分解爲主要決定部分

In this appendix, we show how the forward rate structure is related to the market's rate expectations, bond risk premia and convexity bias. In particular, the holding-period return of an n-year zero-coupon bond can be described as a sum of its horizon return given an unchanged yield curve and the end-of-horizon price change that is caused by a change in the n-1 year constant-maturity spot rate (\(\Delta s_{n-1}\)). The horizon return equals a one-year forward rate, and the end-of-horizon price change can be approximated by duration and convexity effects. These relations are used to decompose near-term expected bond returns and the one-period forward rates into simple building blocks. All rates and returns used in the following equations are compounded annually and expressed in percentage terms.

在本附錄中，咱們展現了遠期收益率結構與市場收益率預期、債券風險溢價和凸度誤差的關係。特別是，n年零息債券的持有期回報能夠被描述爲給定不變收益率曲線的持有期回報與n-1年期即期收益率引發的期末價格變更之和。持有期回報（不變曲線）等於一年期遠期收益率，而期末價格變化能夠由久期和凸度近似。這些關係用於將短時間預期債券回報和一年期遠期收益率分解爲簡單的模塊。如下等式中使用的全部收益率和回報均爲年化並以百分比表示。

\[ \begin{aligned} \frac{h_n}{100} &= \frac{P^*_{n-1} - P_n}{P_n} = \frac{(P^*_{n-1} - P_{n-1}) + (P_{n-1} - P_n)}{P_n} \\ &= (\frac{\Delta P_{n-1}}{P_{n-1}} * \frac{P_{n-1}}{P_n}) + (\frac{P_{n-1}}{P_n} - 1), \end{aligned} \tag{5} \]

where \(h_n\) is the one-period holding-period return of an n-year bond, \(P_n\) is its price (today), \(P_{n-1}^*\) is its price in the next period (when its maturity is n-1), and \(\Delta P_{n-1} = P^*_{n-1} - P_{n-1}\). The second term on the right-hand side of Equation (5) is the bonds rolling yield (horizon return). The first term on the right-hand side of Equation (5) is the instantaneous percentage price change of an n-1 year zero, multiplied by an adjustment term \(P_{n-1}/P_n\).¹²

其中\(h_n\)是n年期債券的一（年）期持有期回報，\(P_n\)爲其當前價格（當下），\(P_{n-1}^*\)爲其下一期（期限爲n-1年）的價格，而且\(\Delta P_{n-1} = P^*_{n-1} - P_{n-1}\)。等式（5）右邊第二項是滾動收益率（持有期回報）。等式（5）右邊的第一項是n-1年零息債券的瞬時百分比價格變化乘以修正項\(P_{n-1}/P_n\)。

Equation (6) shows that the zero's rolling yield (\(P_{n-1} /P_n - 1\)) equals, by construction, the one-year forward rate between n-1 and n. Moreover, the adjustment term equals one plus the forward rate.

等式（6）代表零息債券的滾動收益率（\(P_{n-1} /P_n - 1\)）經過構造等於n-1年期和n期年之間的一年期遠期收益率。此外，修正項等於一加上遠期收益率。

\[ 1+ \frac{f_{n-1,n}}{100} = \frac{(1+\frac{s_n}{100})^n}{(1+\frac{s_{n-1}}{100})^{n-1}} = \frac{P_{n-1}}{P_n}. \tag{6} \]

Equation (7) shows the well-known result that the percentage price change (\(\Delta P / P\)) is closely approximated by the first two terms of a Taylor series expansion, duration and convexity effects.

公式（7）顯示了價格變化百分比能夠由泰勒級數展的前兩項（久期和凸度）近似這一衆所周知的結果。

\[ 100 * \frac{\Delta P}{P} \approx -Dur*(\Delta s) + 0.5 * Cx * (\Delta s)^2, \tag{7} \]

where \(Dur \equiv - \frac{dP}{ds}*\frac{100}{P}\) and \(Cx \equiv \frac{d^2P}{ds^2}*\frac{100}{P}\).

其中\(Dur \equiv - \frac{dP}{ds}*\frac{100}{P}\)，\(Cx \equiv \frac{d^2P}{ds^2}*\frac{100}{P}\)。

Plugging Equations (6) and (7) into (5), we get:

將等式（6）和等式（7）代入等式（5），獲得：

\[ h_n \approx f_{n-1,n} + (1+\frac{f_{n-1,n}}{100})*[-Dur_{n-1}*(\Delta s_{n-1}) + 0.5*Cx_{n-1}*(\Delta s_{n-1})^2]. \tag{8} \]

Even if the yield curve shifts occur during the horizon, for performance calculation purposes the repricing takes place at the end of horizon. This disparity causes various differences between the percentage price changes in Equations (7) and (8). First, the amount of capital that experiences the price change grows to (\(1+f_{n-1,n}/100\)) by the end of horizon. Second, the relevant yield change is the change in the n-1 year constant-maturity rate, not in the n-year zero's own yield (the difference is the rolldown yield change).¹³ Third, the end-of-horizon (as opposed to the current) duration and convexity determine the price change.

即便在持有期收益率曲線發生變化，爲了表現計算須要，從新訂價發生在持有期末。這種差別致使等式（7）和（8）中的價格變更百分比之間的各類差別。首先，經歷價格變化的資本數量在持有期末逐漸增加到\(1+f_{n-1,n}/100\)。第二，相關收益率變化是n-1年期收益率的變化，而不是n年期收益率（差額是下滑收益率變更）。第三，期末（而不是當前）的久期和凸度決定了價格變更。

The realized return can be split into an expected part and an unexpected part. Taking expectations of both sides of Equation (8) gives us the n-year zero's expected return over the next year.

實現的回報能夠分爲預期部分和非預期部分。對等式（8）的兩邊求指望獲得明年的n年期零息債券預期回報。

\[ E(h_n) \approx f_{n-1,n} + (1+\frac{f_{n-1,n}}{100})*[-Dur_{n-1}*E(\Delta s_{n-1}) + 0.5*Cx_{n-1}*E(\Delta s_{n-1})^2]. \tag{9} \]

Recall from Equation (6) that the one-period forward rate equals a zero's rolling yield, which can be split to yield and rolldown return components. In addition, the expected yield change squared is approximately equal to the variance of the yield change or the squared volatility, \(E(\Delta s_{n-1})^2 \approx (Vol(\Delta s_{n-1}))^2\). This relation is exact if the expected yield change is zero. Thus, the zero's near-term expected return can be written (approximately) as a sum of the yield income, the rolldown return, the value of convexity, and the expected capital gains from the rate "view" (see Equation (3)).

回顧等式（6），一期遠期收益率等於零息債券的滾動收益率，能夠被分割爲收益率回報和降低迴報。此外，收益率變化平方的預期近似等於收益率變化方差或波動率平方（\(E(\Delta s_{n-1})^2 \approx (Vol(\Delta s_{n-1}))^2\)）。若是收益率變化的指望爲零，則該關係是準確的。所以，零息債券的近期預期回報能夠寫成收益率回報、下滑回報、凸度價值和從「觀點」獲得的預期資本回報的總和（見公式（3））。

We can interpret the expectations in Equation (9) to refer to the market's rate expectations. Mechanically, the forward rate structure and the market's rate expectations on the right-hand side of Equation (9) determine the near-term expected returns on the left-hand side. These expected returns should equal the required returns that the market demands for various bonds if the market's expectations are internally consistent. These required returns, in turn, depend on factors such as each bond's riskiness and the market's risk aversion level. Thus, it is more appropriate to think that the market participants, in the aggregate, set the bond market prices to be such that given the forward rate structure and the consensus rate expectations, each bond is expected to earn its required return.¹⁴

咱們能夠將等式（9）中的指望解釋爲市場的收益率預期。等式（9）右側的遠期收益率結構和市場收益率預期決定了左邊的近期預期回報。若是市場預期內在一致，這些預期回報應等於市場對各類債券的所要求回報。這些所要求的回報反過來依賴於每一個債券的風險水平和市場風險厭惡程度等因素。所以，更合適的觀點是，市場參與者整體上依據遠期收益率結構和一致收益率預期來設定債券市場價格，每一個債券預期得到所要求的回報。

Subtracting the one-period riskless rate (\(s_1\)) from both sides of Equation (9), we get:

在等式（9）兩端減去一年期無風險收益率（\(s_1\)）獲得：

\[ E(h_n - s_1) \approx f_{n-1,n} -s_1 + (1+\frac{f_{n-1,n}}{100})*[-Dur_{n-1}*E(\Delta s_{n-1}) + 0.5*Cx_{n-1}*(Vol(\Delta s_{n-1}))^2]. \tag{10} \]

We define the bond risk premium as \(BRP_n \equiv E(h_n - s_1)\) and the forward-spot premium as \(FSP_n \equiv f_{n-1,n} - s_1\). The forward-spot premium measures the steepness of the one-year forward rate curve (the difference between each point on the forward rate curve and the first point on that curve) and it is closely related to simpler measures of yield curve steepness. Rearranging Equation (10), we obtain:

咱們將債券風險溢價定義爲\(BRP_n \equiv E(h_n - s_1)\)，遠期-即期溢價定義爲\(FSP_n \equiv f_{n-1,n} - s_1\)。遠期-即期溢價衡量一年期遠期收益率曲線的陡峭程度（遠期收益率曲線上的每一個點與該曲線上的第一點之間的差別），而且與收益率曲線陡峭程度的簡單度量密切相關。從新排列等式（10），獲得：

\[ FSP_n \approx BRP_n + (1+\frac{f_{n-1,n}}{100})*[-Dur_{n-1}*E(\Delta s_{n-1}) + 0.5*Cx_{n-1}*(Vol(\Delta s_{n-1}))^2]. \tag{11} \]

In other words, the forward-spot premium is approximately equal to a sum of the bond risk premium, the impact of rate expectations (expected capital gain/loss caused by the market's rate "view") and the convexity bias (expected capital gain caused by the rate uncertainty). Unfortunately, none of the three components is directly observable.

換句話說，遠期-即期溢價近似等於債券風險溢價、收益率預期的影響（市場收益率「觀點」引發的預期資本損益）與凸度誤差（收益率不肯定性致使的預期資本回報）。不幸的是，這三個組成部分都不能直接觀察到。

The analysis thus far has been very general, based on accounting identities and approximations, not on economic assumptions. Various term structure hypotheses and models differ in their assumptions. Certain simplifying assumptions lead to well-known hypotheses of the term structure behavior by making some terms in Equation (11) equal zero —— although fully specified term structure models require even more specific assumptions. First, if constant-maturity rates follow a random walk, the forward-spot premium mainly reflects the bond risk premium, but also the convexity bias \([E(\Delta s_{n-1}) = 0 \Rightarrow FSP_n \approx BRP_n + CB_{n-1}]\). Second, if the local expectations hypothesis holds (all bonds have the same near-term expected return), the forward-spot premium mainly reflects the market's rate expectations, but also the convexity bias \([BRP_n = 0 \Rightarrow FSP_n \approx Dur_{n-1} * E(\Delta s_{n-1}) + CB_{n-1}]\). Third, if the unbiased expectations hypothesis holds, the forward-spot premium only reflects the market's rate expectations \([BRP_n + CB_{n-1} = 0 \Rightarrow FSP_n \approx Dur_{n-1} * E(\Delta s_{n-1})]\). The last two cases illustrate the distinction between two versions of the pure expectations hypothesis.

迄今爲止的分析很是籠統，基於會計項目和近似值，而不是經濟假設。各類期限結構假說和模型的假設有所不一樣。某些簡化的假設經過使等式（11）中的某些項等於零，致使熟知的期限結構行爲的假設，儘管徹底特定的期限結構模型須要更具體的假設。首先，若是收益率遵循隨機遊走，則遠期-即期溢價主要反映了債券風險溢價，也反映了凸顯誤差。第二，若是局部預期假設成立（全部債券具備相同的近期預期回報），遠期-即期溢價主要反映了市場的收益率預期，也反映了凸顯誤差。第三，若是無偏見的預期假設成立，遠期-即期溢價只反映了市場的收益率預期。最後兩個案例說明了徹底預期假說的兩個版本之間的區別。

APPENDIX B. RELATING VARIOUS STATEMENTS ABOUT FORWARD RATES TO EACH OTHER

附錄B：將遠期收益率的若干結論相互聯繫

In the series Understanding the Yield Curve, we make several statements about forward rates —— describing, interpreting and decomposing them in various ways. The multitude of these statements may be confusing; therefore, we now try to clarify the relationships between them.

在《理解收益率曲線》系列中，咱們以多種方式對遠期收益率進行描述、解釋和分解。如此衆多的陳述可能使人困惑。所以，咱們如今試圖澄清它們之間的關係。

We refer to the spot curve and the forward curves on a given date as if they were unambiguous. In reality, different analysts can produce somewhat different estimates of the spot curve on a given date if they use different curve-fitting techniques or different underlying data (asset universe or pricing source). We acknowledge the importance of these issues —— having good raw material is important to any kind of yield curve analysis —— but in our reports we ignore these differences. We take the estimated spot curve as given and focus on showing how to interpret and use the information in this curve.

咱們考察給定日期的即期收益率曲線和遠期收益率曲線。實際上，使用不一樣的曲線擬合技術或不一樣的底層數據（資產範圍或訂價來源），分析師能夠獲得給定日期即期收益率曲線的不一樣估計。咱們認可這些問題的重要性，良好的原始資料對任何形式的收益率曲線分析都很重要，但在咱們的報告中，咱們忽略了這些差別。咱們按照給定的即期收益率曲線估計結果，重點說明如何解釋和使用該曲線中的信息。

In contrast, the relations between various depictions of the term structure of interest rates (par, spot and forward rate curves) are unambiguous. In particular, once a spot curve has been estimated, any forward rate can be mathematically computed by using Equation (12):

相比之下，收益率期限結構的各類描述（到期、即期和遠期收益率曲線）之間的關係是明確的。特別地，一旦已經估計了一個即期收益率曲線，任何遠期收益率均可以經過使用公式（12）在數學上計算，

\[ (1+\frac{f_{m,n}}{100})^{n-m} = \frac{(1+\frac{s_n}{100})^n}{(1+\frac{s_m}{100})^m}, \tag{12} \]

where \(f_{m,n}\) is the annualized n-m year interest rate m years forward and \(s_n\) and \(s_m\) are the annualized n-year and m-year spot rates, expressed in percent. Thus, a one-to-one mapping exists between forward rates and current spot rates. The statement "the forwards imply rising rates" is equivalent to saying that "the spot curve is upward sloping," and the statement "the forwards imply curve flattening" is equivalent to saying that "the spot curve is concave." Moreover, an unambiguous mapping exists between various types of forward curves, such as the implied spot curve one year forward (\(f_{1,n}\)) and the curve of constant-maturity one-year forward rates (\(f_{n-1,n}\)).

其中\(f_{m,n}\)是m年後的n-m年期收益率，\(s_n\)和\(s_m\)分別是n年期和m年期即期收益率，以百分比表示。所以，遠期收益率和當前即期收益率之間存在一對一的映射。「遠期收益率隱含着收益率上漲」的說法至關於說「即期收益率曲線向上傾斜」，而「遠期收益率隱含着曲線變平」這個說法就至關於說「即期收益率曲線是上凸的」。並且，各類類型的遠期收益率曲線之間存在一個明確的映射關係，好比隱含的一年後即期收益率曲線（\(f_{1,n}\)）和固按期限的一年期遠期收益率曲線（\(f_{n-1,n}\)）。

The forward rate can be the agreed interest rate on an explicitly traded contract, a loan between two future dates. More often, the forward rate is implicitly defined from today's spot curve based on Equation (12). However, arbitrage forces ensure that even the explicitly traded forward rates would equal the implied forward rates and, thus, be consistent with Equation (12). For example, the implied one-year spot rate four years forward (also called the one-year forward rate four years ahead, \(f_{4,5}\)) must be such that the equality \((1+s_5/100)^5 = (1+s_4/100)^4 * (1+f_{4,5}/100)\) holds. If \(f_{4,5}\) is higher than that, arbitrageurs can earn profits by short-selling the five-year zeros and buying the four-year zeros and the one-year forward contracts four years ahead, and vice versa. Such activity should make the equality hold within transaction costs.

遠期收益率能夠是肯定的交易合約的約定收益率，即兩個將來日期之間的貸款收益率。更常見的是根據等式（12），從當下的即期收益率曲線隱含地定義遠期收益率。然而，套利力量確保即便肯定交易的遠期收益率也要等於隱含的遠期收益率，所以與等式（12）一致。例如，隱含的四年後的一年期即期收益率（也稱爲隱含的四年後的一年期遠期收益率，\(f_{4,5}\)）必須確保\((1+s_5/100)^5 = (1+s_4/100)^4 * (1+f_{4,5}/100)\)成立。若是\(f_{4,5}\)高於此值，套利者能夠經過賣出五年期零息債券買入四年期零息債券和約定四年後的一年期遠期合約來賺取利潤，反之亦然。這種套利活動使等式在交易成本容許的範圍內成立。

Forward rates can be viewed in many ways: the arbitrage interpretation; the break-even interpretation; and the rolling yield interpretation. According to the arbitrage interpretation, implied forward rates are such rates that would ensure the absence of riskless arbitrage opportunities between spot contracts (zeros) and forward contracts if the latter were traded. According to the break-even interpretation of forward rates, implied forward rates are such future spot rates that would equate holding-period returns across bond positions. According to the rolling yield interpretation, the one-period forward rates show the one-period horizon returns that various zeros earn if the yield curve remains unchanged. Footnotes 15-17 show that each interpretation is useful for a certain purpose: active view-taking relative to the forwards (break-even); relative value analysis given no yield curve views (rolling yield); and valuation of derivatives (arbitrage).

遠期收益率能夠從多個角度解釋：套利的角度、盈虧平衡的角度和滾動收益率的角度。根據套利角度的解釋，隱含的遠期收益率是確保即期合約（零息債券）和遠期合約（若是後者可交易）之間沒有無風險套利機會的收益率。根據對遠期收益率盈虧平衡角度的解釋，隱含的遠期收益率是確保全部債券頭寸持有期回報相同的將來即期收益率。根據滾動收益率角度的解釋，一（年）期遠期收益率顯示了在收益率曲線保持不變的狀況下，期限零息債券的持有期回報。注15-17顯示，每一種解釋都適用於某一特定目的：相對於遠期收益率，投資者持有本身對收益率的主動觀點（盈虧平衡角度）；假定收益率曲線不變狀況下的相對價值分析（滾動收益率角度）；和衍生品估值（套利角度）。

All of these interpretations hold by construction (from Equation (12)). Thus, they are not inconsistent with each other. For example, the one-period forward rates can be interpreted and used in quite different ways. The implied one-year spot rate four years forward (\(f_{4,5}\)) can be viewed as either the break-even one-year rate four years into the future or the rolling yield of a five-year zero over the next year. Both interpretations follow from the equality \((1+s_5/100)^5 = (1+s_4/100)^4 * (1+f_{4,5}/100)\). This equation shows that the forward rate is the break-even one-year reinvestment rate that would equate the returns between two strategies (holding the five-year zero to maturity versus buying the four-year zero and reinvesting in the one-year zero when the four-year zero matures) over a five-year horizon. Rewriting the equality as \((1+s_4/100)^4 = (1+s_5/100)^5 / (1+f_{4,5}/100)\) gives a slightly different viewpoint; the forward rate also is the break-even selling rate that would equate the returns between two strategies (holding the four-year zero to maturity versus buying the five-year zero and selling it after four years as a one-year zero) over a four-year horizon. Finally, rewriting the equality as \((1+f_{4,5}/100) = (1+s_5/100)^5 / (1+s_4/100)^4\) shows that the forward rate is the horizon return from buying a five-year zero at rate \(s_5\) and selling it one year later, as a four-year zero, at rate \(s_4\) (thus, the constant-maturity four-year rate is unchanged from today). In this series, we focus on the last (rolling yield) interpretation.

全部這些解釋都經過構造（來自等式（12））保持成立。所以，它們彼此並不矛盾。例如，一（年）期遠期收益率能夠用不一樣的方式解釋和使用。隱含的四年後的一年期即期收益率（\(f_{4,5}\)）能夠看做是將來四年保證盈虧平衡的一年期即期收益率，也能夠看做是明年五年期零息債券的滾動收益率。這兩個解釋都遵循等式\((1+s_5/100)^5 = (1+s_4/100)^4 * (1+f_{4,5}/100)\)。這個等式代表，遠期收益率是一個保持盈虧平衡的一年期再投資收益率，能夠保證兩個策略在五年後的回報相等（將五年期零息債券持有到期；購買四年期零息債券，並在四年到期後再投資一年期零息債券）。將等式重寫成\((1+s_4/100)^4 = (1+s_5/100)^5 / (1+f_{4,5}/100)\)，能夠獲得略微不一樣的觀點。遠期收益率也是保持盈虧平衡的賣出收益率，能夠保證兩個策略在五年後的回報相等（將四年期零息債券持有到期；買入五年期零息債券，並在四年後以一年期零息債券賣出）。最後，將等式重寫爲\((1+f_{4,5}/100) = (1+s_5/100)^5 / (1+s_4/100)^4\)，這代表遠期收益率是以\(s_5\)買入五年期零息債券並在一年後以\(s_4\)賣出四年期零息債券的持有期回報（假定固按期限的四年期收益率不變）。在本系列中，咱們重點關注最後一種解釋（滾動收益率角度）。

Interpreting the one-period forward rates as rolling yields enhances our understanding about the relation between the curve of one-year forward rates (\(f_{0,1}, f_{1,2}, f_{2,3}, \dots,f_{n-1,n}\)) and the implied spot curve one year forward (\(f_{1,2}, f_{1,3}, f_{1,4}, \dots,f_{1,n}\)). The latter "break-even" curve shows how much the spot curve needs to shift to cause capital gains/losses that exactly offset initial rolling yield differentials across zeros and, thereby, equalize the holding-period returns. Thus, a steeply upward-sloping curve of one-period forward rates requires, or "implies," a large offsetting increase in the spot curve over the horizon, while a flat curve of one-period forward rates only implies a small "break-even" shift in the spot curve.¹⁵ A similar link exists for the rolling yield differential between a duration-neutral barbell versus bullet and the break-even yield spread change (curve flattening) that is needed to offset the bullet's rolling yield advantage. These examples provide insight as to why an upward-sloping spot curve implies rising rates and why a concave spot curve implies a flattening curve.

將一年期遠期收益率解釋爲滾動收益率，提升了咱們對一年期遠期收益率曲線（\(f_{0,1}, f_{1,2}, f_{2,3}, \dots,f_{n-1,n}\)）與一年後的隱含即期收益率曲線（\(f_{1,2}, f_{1,3}, f_{1,4}, \dots,f_{1,n}\)）之間關係的瞭解。後面的「盈虧平衡」曲線顯示了即期收益率曲線須要變更多少，從而可使產生的資本損益剛好抵消不一樣零息債券的初始滾動收益率差別，進而使持有期回報相等。所以，急劇向上傾斜的一年期遠期收益率曲線要求或者「隱含」，即期收益率曲線在持有期會大幅度上漲，而平坦的一年期遠期收益率曲線僅意味着即期收益率曲線發生小的「盈虧平衡」變化。相似的聯繫存在於久期中性的槓鈴組合與子彈組合之間的滾動收益率差別，而且盈虧平衡的利差變化（曲線變平）用來抵消子彈組合的滾動收益率優點。這些例子說明了「爲何向上傾斜的即期收益率曲線意味着收益率上升」以及「爲何上凸的即期收益率曲線意味着曲線變平」。

Appendix A shows that forward rates can be conceptually decomposed into three main determinants (rate expectations, risk premia, convexity bias). One might hope that the arbitrage, break-even or rolling yield interpretations could help us in backing out the relative roles of rate expectations, risk premia and convexity bias in a given day's forward rate structure. However, such hope is in vain. The three interpretations hold quite generally because of their mathematical nature. Thus, they do not guide us in decomposing the forward rate structure.

附錄A顯示，遠期收益率能夠在概念上分解爲三個主要決定因素（收益率預期、風險溢價、凸度誤差）。人們可能但願基於套利的、盈虧平衡的或滾動收益率的解釋能夠幫助咱們從某一天的遠期收益率結構中推算出收益率預期、風險溢價和凸度誤差的相對做用。可是，這樣的但願是徒勞的。這三種解釋之因此成立整體上而言是基於數學事實。所以，不能指導咱們分解遠期收益率結構。

Therefore, even when two analysts agree that today's forward rate structure is an approximate sum of three components, they may disagree about the relative roles of these components. We can try to address this question empirically. It is closely related to the question about the forward rates' ability to forecast future rate changes and future bond returns. Ignoring convexity bias, if the forwards primarily reflect rate expectations, they should be unbiased predictors of future spot rates (and they should tell little about future bond returns). However, if the forwards mainly reflect required bond risk premia, they should be unbiased predictors of future bond returns (and they should tell little about future rate changes). In Part 2 of this series, Market's Rate Expectations and Forward Rates, we present some empirical evidence indicating that the forward rates are better predictors of future bond returns than of future rate changes.^{16 17}

所以，即便兩位分析師都贊成當下的遠期收益率結構是三個組成部分的總和，他們也可能就這些組成部分的相對做用產生分歧。咱們能夠嘗試用經驗來解決這個問題，這與遠期收益率預測將來收益率變更和將來債券回報的能力密切相關。忽略凸度誤差，若是遠期收益率主要了收益率預期，則遠期收益率應該是將來即期收益率的無偏預測（而且應該對將來的債券回報不提供信息）。然而，若是遠期收益率主要反映債券風險溢價，則遠期收益率應該是將來債券回報的無偏預測（而且應該對將來的收益率變更不提供信息）。在本系列第2部分——《市場收益率預期與遠期收益率》，咱們提出一些實證證據代表，遠期收益率是將來債券回報的預測指標，而不是將來收益率變更。

Finally, our analysis does not reveal the fundamental economic determinants of the required risk premia or the market's rate expectations —— nor does it tell us to what extent the nominal rate expectations reflect expected inflation and expected real rates. Macroeconomic news about economic growth, inflation rates, budget deficits, and so on, can influence both the required risk premia and the market's rate expectations. More work is clearly needed to improve our understanding about the mechanisms of these influences.

最後，咱們的分析沒有揭示風險溢價或市場收益率預期的根本經濟決定因素，也沒有告訴咱們名義收益率預期在多大程度上反映了預期通貨膨脹率和預期實際收益率。關於經濟增加、通脹率、預算赤字等的宏觀經濟新聞能夠影響風險溢價和市場的收益率預期。顯然須要更多的工做來提升咱們對這些影響機制的理解。

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1. The bond risk premium is defined as a bond's expected (near-term) holding-period return in excess of the riskless short rate. Historical experience suggests that long-term bonds command some risk premium because of their greater perceived riskiness. However our term "bond risk premium" also covers required return differentials across bonds that are caused by other factors than risk, such as liquidity differences, supply effects or market sentiment.
   債券風險溢價定義爲債券預期（近期）持有期回報超過無風險短時間回報的部分。歷史經驗代表，長期債券因爲其較高的風險性而具備必定的風險溢價。然而，咱們使用的「債券風險溢價」一詞也涵蓋了由風險之外的其餘因素致使的債券回報差別，如流動性差別、供應效應或市場情緒等。↩

2. A concave (but upward-sloping) curve has a sleeper slope at short maturities than at long maturities: thus, a line connecting two points on the curve is always below the curve. A convex (but upward-sloping) curve has a steeper slope at long maturities than at short maturities: thus, a line connecting two points on the curve is always above the curve.
   上凸（可是向上傾斜的）曲線在短時間端的陡峭程度比長期端要大：所以，鏈接曲線上兩點的直線老是在曲線下方。一個下凸（可是向上傾斜的）曲線在長期端的陡峭程度比短時間端要大：所以，鏈接曲線上兩點的直線老是在曲線上方。↩

3. All rates and returns in this report are expressed in percentage terms (200 basis points = 2%), whereas in the
   equations in Parts 1 and 2 of this series they were expressed in decimal terms (200 basis points = 0.02).
   本報告中的全部收益率和回報均以百分比表示（200基點 = 2%），而在這個系列的第1、二部分中的等式，用十進制表示（200基點 = 0.02）。↩

4. We hasten to point out that these calculations are quite imprecise, especially at long durations. Even an error of a couple of basis points in our proxy for the market's rate expectation will have a large impact on any long bond's expected return (and thus on the estimated bond risk premium), because the expected yield change is scaled up by duration. Such sensitivity reduces the usefulness of this decomposition at long durations.
   咱們指出這些計算至關不許確，特別是在長久期端。即便市場收益率預期的指代只有幾個基點的偏差，也會對任何長期債券的預期回報（以及估計的債券風險溢價）產生巨大的影響，由於預期收益率變化會隨着久期的推移而放大。這種敏感性會下降長久期端分解的有效性。↩

5. In the earlier parts of this series, we provide further evidence of the importance of time-varying risk premia. Why do so many market participants and analysts think that the rate expectations are much more important determinants of the yield curve steepness than are the bond risk premia, in spite of the contradicting empirical evidence? We offer one possible explanation: Individual market participants have their individual rate views and individual required risk premia. (Few investors think explicitly in terms of such premia, but they will extend duration only if they expect longer bonds to outperform shorter bonds by a margin sufficient to offset their greater risks.) However, what matters for the yield curve is the market's aggregate rate view and aggregate required risk premia. Both vary over time as individual rate views and risk perceptions and preferences change (and its the composition of market participants changes). We stress that the changing individual rate view may have smaller aggregate effects than the changing risk perceptions and preferences even if individual rate views are more volatile than individual risk perceptions and preferences. This effect can occur if the changes in risk perceptions and preferences are much more highly correlated across individuals than are changes in rate views. For example, when volatility is high, most market participants are likely to demand abnormally high bond risk premia even if they have widely different views about the future rate direction. Perhaps most market participants focus on the fact that (their) individual rate views vary more over time than do their risk perceptions and preferences, ignoring the fact that market rates are driven by the aggregate effects, for which correlations across individuals matter a lot.
   在本系列前面的部分，咱們提供了時變風險溢價重要性的進一步證據。爲何儘管有相互矛盾的經驗證據，不少市場參與者和分析師依然認爲收益率預期對收益率曲線陡峭程度的決定性比債券風險溢價要重要得多？咱們提供了一個可能的解釋：個別市場參與者有他們本身的收益率觀點和我的要求的風險溢價。（不多有投資者認爲這種溢價是明確的，可是隻有當他們預期長期債券表現超太短期債券的幅度足以抵消更大的風險時，他們纔會增長久期。）然而，對收益率曲線重要的是市場的整體收益率觀點和整體要求的風險溢價。隨着我的收益率觀點與風險認知和偏好的變化（及其市場參與者的組成變化），二者都隨時間而變化。咱們強調，即便我的收益率觀點比我的風險認知和偏好更具波動性，變化的我的收益率觀點的整體效應也可能小於變化的風險認知和偏好。若是風險認知和偏好的變化在我的之間的相關度高於收益率率觀點的變化，則可能會發生這種影響。例如，當波動率較高時，大多數市場參與者即便對將來的收益率走向有不一樣的見解，也可能要求異常高的債券風險溢價。也許大多數市場參與者關注的事實是，他們我的的收益率觀點隨着時間的推移而變化，而不是他們的風險認知和偏好，忽略了市場收益率是由整體效應驅動的事實（我的之間的相關度對整體效應影響很大）。↩

6. However, certain modifications are needed when Equation (3) is used to describe the expected returns of coupon bonds rather than those of zeros —— and the approximation will be somewhat worse. We use each bond's rolling yield to measure the horizon return given an unchanged yield curve; this measure no longer equals the one-period forward rate. We also use the end-of-horizon duration and convexity as well as the change in the constant-maturity rate of a constant-coupon curve at horizon, and we adjust the duration and convexity effects for the fact that the bonds value increases to (1 + rolling yield / 100) by the end of the horizon. Besides the approximation error of ignoring higher-order terms than duration and convexity effects, another source of error exists for coupon bonds: The reinvestment rate assumptions vary across bonds. Recall that the calculation of the yield to maturity implicitly assumes that all cash flows are reinvested at the bond's yield to maturity. This fact may lead to exaggerated estimates of yield income for long-term bonds if the yield curve is upward sloping, a problem common to all expected return measures that use the concept of yield to maturity. Even though our approach of using bond-specific yields does not ensure internal consistency of the reinvestment rate assumptions across bonds, any inconsistencies should have a relatively small impact on the overall of bonds' expected returns.
   然而，當等式（3）被用來描述付息債券而不是零息債券的預期回報時，須要進行某些修改，並且近似值會稍差一些。咱們使用每一個債券的滾動收益率來度量收益率曲線不變狀況下的持有期回報，這一度量再也不等於一年期遠期收益率。咱們利用期末的久期和凸度以及固息曲線在固按期限收益率變化，並調整久期和凸度效應，由於債券的價值在上升期末到（ 1 + 滾動收益率/100）。除了忽略久期和凸度以外的高階項形成度的逼近偏差，對於付息債券還有另一個的偏差來源：假設再投資率在債券之間是不一樣的。回想一下，到期收益率的計算暗含着全部的現金流量都是在債券到期收益率的基礎上再投資的。若是收益率曲線向上傾斜，那麼這個事實可能會致使對長期債券收益率回報的誇大估計，這是全部使用到期收益率進行預期回報度量的共同問題。儘管咱們使用債券特定收益率的方法並不能確保債券再投資率假設的內部一致性，但任何不一致對債券預期回報的整體影響應該是相對較小的。↩

7. The yield to maturity of a single cash flow is unambiguous, whereas the yield of a portfolio of multiple cash flows is a more controversial measure. The duration-times-market-value weighted yield is a good proxy for a portfolio's true yield to maturity (internal rate of return). However, a portfolio's market-value weighted yield may be a better estimate of the portfolio's likely yield income over a short horizon (its near-term expected return) than is its yield to maturity. The yield to maturity weighs longer cash flows more heavily and is more influenced by the built-in reinvestment rate assumptions. We will return to this topic in a future report.
   單一現金流量的到期收益率是明確的，而多個現金流量組合的收益率是一個更有爭議的度量方式。以久期乘市值加權的收益率是投資組合真實到期收益率（內部收益率）的良好指標。然而，投資組合的市值加權收益率多是短時間內投資組合的收益率回報（其近期預期回報）更好的估計值，而不是其到期收益率。到期收益率在長期現金流上的權重更大，而且受內在再投資率假設的影響更大。咱們將在將來的報告中回到這個話題。↩

8. Whether such local cheapness effects appear as deviations from a fitted yield curve or as "wiggles" or "kinks" in the fitted curve depends on the curve-estimation technique. Recall that all curve-estimation techniques try to fit bond prices well while keeping the curve reasonably shaped. If the goodness of fit is heavily weighted, all bonds have small or no deviations from the fitted curve. However, a close fit may lead to "unreasonably" jagged forward rate curves. Based on Equation (2), the forward rate curve should be smooth, rather than jagged because maturity-specific expectations of rate or volatility behavior are hard to justify and because arbitrageurs presumably are quick to exploit any abnormally large expected return differentials between adjacent-maturity bonds.
   這種局部低估效應是否表現爲與擬合收益率曲線的誤差或擬合曲線中的「擺動」或「扭結」取決於曲線估計技術。回想一下，全部的曲線估計技術都試圖在保持曲線合理形狀的前提下同時很好地擬合債券價格。若是擬合優度的權重很大，則全部債券與擬合曲線的誤差都很小或沒有誤差。然而，完美的擬合可能致使「不合理」的良莠不齊的遠期收益率曲線。基於等式（2），遠期收益率曲線應該是平滑的，而不是良莠不齊的，由於期限特定的收益率預期或波動率行爲難以成立，由於套利者可能很快會利用相鄰期限之間任何異常大的預期回報差別。↩

9. In our analysis, we include the local effects into the expected bond returns separately as a fifth term. As an alternative, we could include the financing advantage (repo income) and the spread off the curve in the yield income, and we could include the expected cheapening in the rolldown return. "Rich" bonds, such as the on-the-runs, are unlikely to roll down the fitted curve if the overall curve remains unchanged. More likely, they will lose their relative richness eventually. It may be reasonable to assume that an on-the-run bond's yield advantage and expected cheapening exactly offset its expected financing advantage.
   在咱們的分析中，咱們將局部效應做爲第五項歸入債券預期回報。做爲其餘選擇，咱們能夠將融資優點（回購收益）和相對與曲線的利差歸入到收益率回報，並且咱們能夠在下滑回報中歸入預期的低估量。若是總體曲線保持不變，那麼「貴的」債券，例如活躍券，就不太可能在擬合曲線上下滑（譯註：收益率減少）。更可能的是，他們最終會失去相對的高估值。能夠合理地假設，一個活躍券的收益率優點和預期的低估量剛好抵消了其預期融資優點。↩

10. Realized returns can be split into an expected part and an unexpected part, and both parts can be decomposed further. Equation (3) describes the decomposition of the expected part, while the unexpected part can be split into duration and convexity effects. This type of return attribution can have a useful role in risk management and performance evaluation, but these two activities are not our focus in this report.
    實現的回報能夠分爲預期部分和非預期部分，而且這兩部分能夠進一步分解。等式（3）描述了預期部分的分解，而非預期部分能夠分解爲久期和凸度效應。這種類型的回報歸因在風險管理和績效評估中是有用的工具，可是這兩個課題並非咱們這份報告關注的。↩

11. It is possible to compute the value of convexity and the duration impact in another way and end up with almost exactly the same numbers. If we compute the bond returns based on yield curve scenarios that are, on average, unbiased or "viewless" (that is, we subtract the ten-basis-point mean yield change from each scenario), the probability-weighted expected portfolio return is 7.02%. We can estimate the duration impact of a view by comparing the portfolio's probability-weighted expected return given the five scenarios with the rate view (6.82%) to the expected return given the "viewless" set of scenarios. The difference is again -20 basis points. Similarly, we can estimate the value of convexity by comparing the probability-weighted expected return of the portfolio under a "viewless" set of scenarios (7.02%) to the expected return under the neutral scenario that has the same average rate view —— an unchanged yield curve —— but assumes no volatility (7.00%). The difference, two basis points, measures the pure effect of the implicit volatility forecast without a bias from a yield curve view.
    能夠以另外一種方式計算凸度價值和久期影響，最後獲得幾乎徹底相同的數字。若是咱們基於平均來看無偏或「無觀點」的收益率曲線情景（即從每一個情景中減去10個基點的平均收益率變化）來計算債券回報，那麼機率加權的預期投資組合回報是7.02%。咱們能夠經過比較給定收益率觀點情境下的投資組合機率加權預期回報（6.82%）與給定「無觀點」情景下的預期回報，來估計「觀點」的久期影響。差額依然是-20個基點。一樣，咱們能夠經過比較在「無觀點」情景下的投資組合機率加權預期回報（7.02%）與中性情景下（具備相同的平均收益率觀點但假設沒有波動率，收益率曲線不變）的預期回報（7.00%），來估計凸度價值。兩個基點的差別衡量了徹底屬於隱含波動率預測的效應，而沒有收益率曲線觀點帶來的誤差。↩

12. The adjustment term is needed because the bond's instantaneous price change occurs at the end of horizon, not today. The value of the bond position grows from one to \(P_{n-1}/P_n\) at the end of horizon if the yield curve is unchanged. The end-of-horizon value \(P_{n-1}/P_n\) would be subject to the yield shift at horizon.
    調整項是必要的，由於債券的瞬時價格變化發生在持有期末，而不是當前。若是收益率曲線不變，則債券頭寸的價值從1增加到持有期末的\(P_{n-1}/P_n\)。持有期末的價值\(P_{n-1}/P_n\)將受到持有期內收益率變更的影響。↩

13. If we used bonds' own yield changes in Equation (8), these yield changes would include the rolldown yield change. In this case, we should not use the forward rate (which includes the impact of the rolldown yield change on the return, in addition to the yield income) as the first term on the right-hand side of Equation (8). Instead, we would use the spot rate.
    若是咱們在等式（8）中使用債券自己的收益率變化，這些收益率變化將包括下滑收益率的變化。在這種狀況下，咱們不該該把遠期收益率（此時除了收益率回報外還包含了下滑收益率變化對回報的影響）做爲等式（8）右邊的第一項。相反，咱們會使用即期收益率。↩

14. Individual investors also can use Equation (9) but the interpretation is slightly different because most of their investments are so small that they do not influence the market rates; thus, they are "price-takers". Any individual investor can plug his subjective rate expectations into Equation (9) and back out the expected return given these expectations and the market-determined forward rates. These expected returns may differ from the required returns that the market demands: this discrepancy may prompt the investor to trade on his view.
    我的投資者也可使用等式（9），但其解釋略有不一樣，由於他們的大部分投資很是之小，以致於不影響市場收益率，他們是「價格接受者」。 任何我的投資者均可以將他的主觀收益率預期歸入方程（9），在給定這些預期和市場決定的遠期收益率的狀況下推算出預期回報。這些預期回報可能不一樣於市場所要求的回報：這種差別可能促使投資者按照他的觀點進行交易。↩

15. Part 1 of this series describes one common way to use the break-even forward rates. Investors can compare their subjective views about the yield curve at some future date (or about the path of some constant-maturity rate over time) to the forward rates and directly determine whether bullish or bearish strategies are appropriate. If the rate changes that the forwards imply are realized, all bonds earn the riskless return because \((1 + s_n/100)^2 / (1 + f_{1,n}/100)^{n-1} = 1 + s_1/100\). If rates rise by more than that, long bonds underperform short bonds. If rates rise by less than that, long bonds outperform short bonds (because their capital losses do not quite offset their initial yield advantage).
    本系列的第1部分介紹了使用盈虧平衡遠期收益率的一種常見方法。投資者能夠將他們對將來某個時間收益率曲線的主觀觀點（或者某個固按期限收益率隨時間推移的路徑）與遠期收益率進行比較，直接肯定看漲或看跌策略是否合適。若是遠期收益率隱含的變化最終實現，則全部債券都得到無風險回報，由於\((1 + s_n/100)^2 / (1 + f_{1,n}/100)^{n-1} = 1 + s_1/100\)。 若是收益率上漲超過那些量，長期債券的表現將不如短時間債券。若是收益率上升的幅度小於那些量，長期債券的表現則優於短時間債券（由於它們的資本損失並不能徹底抵消其最初的收益率優點）。↩

16. This evidence also suggests that the current spot curve is a better predictor of the next-period spot curve than is the implied spot curve one period forward. These findings imply that the rolling yields are reasonable proxies for the near-term expected bond returns —— although even rolling yields capture a very small part of the short-term realized bond returns. Note that the poorer the forwards are in predicting future rate changes, the better they are in predicting bond returns —— because the the implied rate changes that would offset initial yield advantages tend to occur more rarely. Note also that some investors may not care whether the forwards' ability to predict bond returns reflects rational risk premia or the market's inability to forecast rate changes: they want to earn any predictable profit irrespective of its reason.
    這個證據還代表，相對於一（年）期後隱含的即期曲線，當前的即期曲線是下一（年）期即期曲線更好的預測指標。這些發現意味着滾動收益率是近期債券預期回報的合理替代指標——儘管滾動收益率只能捕捉短時間實現債券回報的很小一部分。請注意，遠期收益率預測將來收益率變化的效果越差，在預測債券回報方面就越好——由於抵消初始收益率優點的隱含收益率變更更少發生。還要注意的是，有些投資者可能並不在意遠期收益率預測債券回報的能力是否反映了合理的風險溢價或者市場沒法預測收益率變化：他們想要得到任何可預測的利潤，而無論其緣由。↩

17. One common misconception is that the forward rates are used in the valuation of swaps, options and other derivative instruments because the forwards are good predictors of future spot rates. In fact, the forwards' ability to predict future spot rates has nothing to do with their usefulness in derivatives pricing. Unlike forecasting returns the valuation of derivatives is based on arbitrage arguments. For example, traders can theoretically construct, by dynamic hedging, a riskless combination of a risky long-term bond and an option written on it. The price of the option should be such that the hedged position earns the riskless rate —— otherwise a riskless arbitrage opportunity arises. The forward rates are central in this valuation because the traders via their hedging activity, can lock in these rates for future periods. This arbitrage argument implies that the yield curve option pricing models should be calibrated to be consistent with the market forward rates despite the fact that the forwards are quite poor predictors of future spot rates.
    一個常見的誤解是遠期收益率被用於互換、期權和其餘衍生品的估值，由於遠期收益率是將來即期收益率的良好預測指標。事實上，遠期收益率預測將來即期收益率的能力與其在衍生產品訂價中的用處無關。與預測回報不一樣，衍生品的估值是基於套利的觀點。例如，交易者能夠經過動態對衝，在理論上將長期債券和對應的期權構造出無風險的組合。期權的價格應該保證被對衝頭寸得到無風險收益率——不然就會產生無風險的套利機會。遠期收益率是估值的核心，由於交易者經過對衝活動能夠在將來鎖定這些收益率。這種套利的觀點意味着收益率曲線期權訂價模型應該校準到與市場遠期收益率一致，儘管遠期收益率對將來即期收益率的預測是至關差的。↩